I once caught an 11th-grader who snuck a cheat sheet into the final exam.
At first, he tried to shuffle it under some scratch paper. When I cornered him, he shifted tactics. “It’s my page of equations,” he told me. “Aren’t we allowed a formula sheet? The physics teacher lets us.” Nice try, but no dice. The principal and I rejected his alibi and hung a fat zero on his final exam. That dropped his precalculus grade down from a B+ to a D+. It lingered like a purple bruise on his college applications.
Looking back, I have to ask myself: Why didn’t I allow a formula sheet? Cheat sheets aim to substitute for memorization, and I hate it when my students memorize things.
"What’s the sine of π/2?" I asked my first-ever trigonometry class.
"One!" they replied in unison. "We learned that last year."
So I skipped ahead, later to realize that they didn’t really know what “sine” even meant. They’d simply memorized that fact. To them, math wasn’t a process of logical discovery and thoughtful exploration. It was a call-and-response game. Trigonometry was just a collection of non-rhyming lyrics to the lamest sing-along ever.
Read more. [Image: Amy Loves Yah/Flickr]
The Beautiful Blackboards of Quantum Physics Labs
Alejandro Guijarro visited the world’s finest quantum physics labs to record their half-erased blackboards. In an era where science is increasingly all-digital, it’s a striking reminder that science is still an active, hands-on process, a process here captured in layers of smeared chalk.
The orbits of the moons and planets form a 4-dimensional fractal helix in spacetime.
Because it’s that time of year again!!
Also…this is a reblog of one of my very first posts on Proof! :)
The Von Koch snowflake!
The Von Koch snowflake is a fractal which is constructed from an equilateral triangle as follows:
1) remove the middle third of each side,
2) build a new equilateral triangle on each of the resulting gaps,
3) repeat steps 1 and 2 for the new object.
If you keep going indefinitely, you end up with the fractal shown on the right.
The outline of the snowflake is incredibly crinkly. In fact, it does not contain any straight line pieces at all: the middle third of any piece of straight line has been replaced by two sides of a triangle, creating a spike. Mathematicians measure the crinkliness of a fractal by the fractal dimension, a generalisation of our ordinary notion of dimension. The outline of our snowflake is too crinkly to be one-dimensional. On the other hand, it clearly is not two-dimensional either, since it contains no area. In this case, the dimension lies between 1 and 2, in fact it is equal to log(4)/log(3) = 1.2619.