I'm a mathematician, and I actually hadn't seen the proof that $A = \pi r^2$ via showing that the shape formed by that particular arrangement of the slices of the circle becomes a square as the number of slices is taken to infinity. That's a very cool proof.
Out of curiosity, were you perhaps taught with some other seemingly-intuitive explanation? The one given here was the first (and, really, only) such explanation that I received in grade school.
It was also my first real exposure to the concept of infinity and limits in math class (although my brother had stumbled upon and explained to me the concept of a derivative in his head one day while sitting in church, way before either of us would be exposed to calculus).
He didn't prove the wiggly-sided paralellogram -> rectangle transition when the angle of each slice goes to 0. He also did not prove that he is allowed to take the angle to 0. I can grant him that any angle > 0 results in a less and less squigglier paralellogram with more and more vertical sides with an obvious proof, but that doesn't prove either that the limit is a rectangle, or that reasoning at the limit is transitive to reasoning about the circle.
I don't doubt the result or the method - they both seem intuitively correct, they just aren't proven.
I really love how mathematics takes much more sense when explained this way, with historical background. It gives the sense of a story, of people struggling with concepts and generations needed to solve them, not only the area but also how they didn't have calculus, all this parts they make easier for me to understand concepts and remember concepts. When you put 2000 years of mathematic development in front of a person and explain it like it was obvious, something brakes inside you.
I understand that from a certain point you have to study mathematics the hard way. But for kids and people starting with mathematics, it´s better a "history-story" approach.
I don't know whether the actual history of mathematics is useful for understanding mathematics.
However, I do believe that a plausible story of how somebody came up with concepts is very important. It can be tempting to structure a mathematics lecture in a very sober/bland style of definitions followed by theorems with proof. However, definitions usually arose because somebody was trying to solve a particular problem, and this should be part of mathematics teaching.
It is usually said that "mathematics is not a spectator sport". To fully learn it, you have to put yourself into the shoes of somebody who invents certain definitions and seeks out certain theorems because they have some ulterior question that motivates them.
I would go beyond what you wrote by saying that this applies to advanced mathematics as well. The truly great papers tend to lucidly lay out the thought processes that explain why definitions and proof steps are chosen to be of a certain form.
For me it is. For example, I read "Fermat´s last theorem". The great thing is that the author tells in a vivid way, how different mathematicians attacked different parts of the theorem, starting with Fermat, up to the present
. He also explains how the tools developed along the time found real world applications.
I can not say that I understand the concepts explained at the book, but now I feel that I could even start learning Number theory (well maybe this is a stretch, but you get the idea)
On a side note please for give my parent comment. I was writing from the Iphone seated at a plane, so I had to hurry and didn't have time for corrections.
If you're interested in pi, the book A History of Pi by Petr Beckman is a must read, not only for the good information but also for his nonconventional approach.
Note that Archimedes used a 96-sided polygon to estimate bounds on the value of pi (http://en.wikipedia.org/wiki/Pi)! This was quite a feat since arithmetic calculations and algebraic was one area that ancient Greeks were weak at.
If you're fascinated by infinity, you should check out 'Infinity and the mind' by Rudy Rucker. It really puts the concept of infinity in it's historical perspective and even contains a unique conversation between Rucker and Godel. (shameless plug: I wrote a review on my blog http://branchandbound.net/blog/bookreview/2013/01/bookreview...)
Also, there is a very good, popular movie by BBC Four 'Dangerous Knowledge' [0].
"Beneath the surface of the world, are the rules of science. But beneath them, there is a far deeper set of rules – a matrix of pure mathematics which explains the nature of the rules of science and how it is way we can understand them in the first place. In this one-off documentary, David Malone looks at four brilliant mathematicians – Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing – whose genius has profoundly affected us, but which tragically drove them insane and eventually led to them all committing suicide.
The film begins with Georg Cantor, the great mathematician whose work proved to be the foundation for much of the 20th-century mathematics. He believed he was God’s messenger and was eventually driven insane trying to prove his theories of infinity."
I beg to differ - this "documentary" is complete crap.
Some of the math is acceptable, some of it is atrocious, and the assertions that thinking about infinity is what drove these men insane is just completely bonkers.
