I was recently turned on to an absolutely wonderful math and science educator whose videos would do well to provide the backbone of the next generation of the Common Core, at least in math education. Vi Hart is a videographer (is that what one would call someone who makes videos) who specializes in a playful, though very serious approach to expose and illustrate complicated science concepts and techniques. One of her series, entitled Doodling in Math Class, exposes the rich, playful and beautiful structure inherent in every math class but lost in the tedium of sterile, and solely utile function. Below is a three part video explaining why and how the Fibonacci Sequence (not a series, really) shows up so often in nature. It is mind-boggingly well done, IMHO:
The other two parts follow immediately from this one. Give them a look!
BTW, THIS is what mathematics is really about. Vi's money quote (at the end of the third part):
This is why science and math are so much fun. You discover things that seem impossible to be true, and then you get to figure out why it is impossible for them not to be true.