“…if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done — I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.”
Using calculus, we can ask all sorts of questions:
Algebra & calculus are a problem-solving duo: calculus finds new equations, and algebra solves them. Like evolution, calculus expands your understanding of how Nature works.
An Example, Please
- How does an equation grow and shrink? Accumulate over time?
- When does it reach its highest/lowest point?
- How do we use variables that are constantly changing? (Heat, motion, populations, …).
- And much, much more!
Now here’s where things get funky. Let’s unroll those rings and line them up. What happens?
Neural learning algorithms adaptively update their parameters using gradient descent and backpropagation to minimize prediction errors- We get a bunch of lines, making a jagged triangle. But if we take thinner rings, that triangle becomes less jagged (more on this in future articles). One side has the smallest ring (0) and the other side has the largest ring ( ) We have rings going from radius 0 to up to “r”. For each possible radius (0 to r), we just place the unrolled ring at that location. The total area of the “ring triangle” = , which is the formula for area!
- Integral of Sin(x): Geometric Intuition
I have a love/hate relationship with calculus: it demonstrates the beauty of math and the agony of math education.
Calculus relates topics in an elegant, brain-bending manner. My closest analogy is Darwin’s Theory of Evolution: once understood, you start seeing Nature in terms of survival. You understand why drugs lead to resistant germs (survival of the fittest). You know why sugar and fat taste sweet (encourage consumption of high-calorie foods in times of scarcity). It all fits together.
Calculus is similarly enlightening. Don’t these formulas seem related in some way?
- We get a bunch of lines, making a jagged triangle. But if we take thinner rings, that triangle becomes less jagged (more on this in future articles).
- One side has the smallest ring (0) and the other side has the largest ring (
- We have rings going from radius 0 to up to “r”. For each possible radius (0 to r), we just place the unrolled ring at that location.
- The total area of the “ring triangle” =
