Be Afraid, Be Very Very Afraid

Two for the price of one today:

1)  First, in time for Halloween, DO NOT miss the frightful tale of Differentiation... as only Ben Orlin can tell it (bwaaahaaaahaaaaa):
http://mathwithbaddrawings.com/2015/10/28/the-differentiation-a-survivors-tale/


2)  Less scary, but more mind-racking perhaps than differentiation, is the 'Sleeping Beauty Problem/Paradox,' which I haven't mentioned for awhile, but do now (...at least one version of it):



The correct answer is: 1/2, 1/3?; 1/2, 1/3?; 1/2 or 1/3???.... two different logical answers, splitting the mind in two, with no final resolution. Spine-tingling stuff! ;-)
My original post on it was back in 2012 with a number of additional links:
http://math-frolic.blogspot.com/2012/03/sleeping-beautynot-your-childhood-fairy.html

Also, Tanya Khovanova had lengthy previous discussion of it on her blog here:
http://blog.tanyakhovanova.com/2011/08/the-sleeping-beauty-problem/

And even physicist Sean Carroll covered it a year ago, drawing 240+ comments:
http://www.preposterousuniverse.com/blog/2014/07/28/quantum-sleeping-beauty-and-the-multiverse/

Pick your side... you'll find some good arguments (and thinkers) backing you up either way.  Spooky indeed!

Calculus Going Viral?… Could It Be

Forbes Magazine reports on Ohio State professor Jim Fowler taking calculus to the Web via Coursera (and getting rave reviews):
http://tinyurl.com/lz2smt2

And he's on YouTube, if you don't want to sign up for Coursera without getting a sampling first:
https://www.youtube.com/user/kisonecat

here's his intro to the course:

Books, Links, Calculus...

I'm busily putting final touches on a long review of Martin Gardner's autobiography [now up HERE], while simultaneously reading Edward Frenkel's new "Love and Math" and Raymond Smullyan's "The Gödelian Puzzle Book" -- don't know if I'll write reviews of either, but have no hesitation recommending both. The latter is typical 'Ray Smullyan,' and perhaps his best, most focused attempt yet to elucidate Gödel's theorems via paradoxical puzzles. The Frenkel book thus far looks wonderful (deep, yet accessible) and I hope will confer (to my naive brain) some sense of what the cutting edge Langlands Program is all about.
For now, I just have time to pass along to readers a few miscellaneous links I've enjoyed the last few days:

1) Just today, the same Edward Frenkel had an interesting piece in Slate on NSA and their 'backdoor' cryptography practices:

http://tinyurl.com/jvq87n6

2) Another fun read from Simon Singh (on The Simpsons' comedy writers), promoting his new book:

http://tinyurl.com/md2x5sz

3) And finally, recently Steven Strogatz enthusiastically tweeted a link to this intro for Robert Ghrist's 1st-year calculus course (from Coursera):

https://www.youtube.com/watch?v=vyYgt_qHYrM

The reason I provide this link at all is because quite awhile back an emailer asked me to recommend a video site on the Web for learning first-year calculus, and even though I'm aware of many, I wasn't confident endorsing any particular one. When I asked readers if they had definite recommendations I got no response. But if Steven Strogatz is willing to give a thumbs-up to this one, I trust his judgment!
One thing that makes it look interesting, beyond the quality of graphics put into it, is Ghrist's unconventional use of Taylor series (which usually come at the end of 1st year calculus), near the beginning of the course.

Tuesday Grab Bag...

A little Newton, Leibniz, Tanton, and Devlin today….

First, a link to James Tanton's latest "Cool Math" newsletter here (pdf):

http://www.jamestanton.com/wp-content/uploads/2012/03/Cool-Math-Newsletter_December2012.pdf

…and if you enjoy that, you better go check out his links to his past newsletters (a treasure-trove of math enlightenment!):

http://www.jamestanton.com/?p=1072

Second, a fun piece on the historical rivalry between Newton and Leibniz over the discovery of calculus:

http://www.mathforgrownups.com/en-garde-the-great-calculus-duel/

a few lines therefrom:

"So Newton farts around with this idea of fluxions, finally getting around to publishing Method of Fluxions in 1736… he published a few manuscripts on the subject, sending early copies to some colleagues. Meanwhile, in Germany, Leibniz was jotting down his own discoveries in his journal. In 1675, he noodled around, finding the area under a the graph of y = f(x) using integral calculus.

"In other words, the two men were discovering calculus at the same time and in completely different parts of the world. (Okay, Germany and England weren’t too distant from one another, but in the 17th century, they may as well have been on different planets.)"...

"...it was neither Newton nor Leibniz who lit the fire of the great calculus war. In 1704, an anonymous review of Newton’s fluxions suggested that he borrowed [i.e. stole] the idea from Leibniz, which of course infuriated Newton. Letters flew back and forth between the two mathematicians and their surrogates."

And finally another great post from Keith Devlin on his experience with the recent math MOOC course (massive open online course) which he instructed:

http://mooctalk.org/2012/11/26/coming-up-for-air-and-spouting-off/

Devlin clearly feels MOOCs are here to stay and be a significant part of future education, but also recognizes the problems they entail. Occasionally when reading him I wonder if we're taking 3 steps forward and 4 steps back… I DON'T really think so (neither does he) but I do pause to wonder...

40 years ago I first learned of behaviorist-based "programmed learning," and thought it too would absolutely revolutionize education. It did nothing of the sort. What Dr. Devlin so well elucidates is the importance of a "social" component to learning (one of the things I think 'programmed learning' lacked). Learning/teaching are not simple uni-directional phenomena, and a 'human touch,' so to speak, is still needed.
Interestingly (since so many prematurely critiqued the Web as an isolating and dehumanizing influence) many now note that the Web or digital age has entered a "social" phase with emphasis on: collaboration, peer-to-peer contact, hive-mind, crowd-sourcing, open-access, and the like (in general, a great break-down of prior barriers to communication).
What I think will insure the progress and future of MOOCs and digital education generally, is its universality and availability to all with internet access (eventually, most everyone worldwide), leveling education opportunities as they have never been leveled before, and finally permitting people to learn at their own pace, in highly individualized ways.
Anyway, to close out, a bit from Dr. Devlin's piece:

"...(in most disciplines) the key to real learning has always been bi-directional human-human interaction (even better in some cases, multi-directional, multi-person interaction), not unidirectional instruction…
"For the vast majority of students, discussion with (and getting feedback from) professors, TAs, and other students struggling to acquire problem solving ability and master abstract concepts and proofs, is an essential part of learning. For those purposes, the online version does not find its inspiration in Khan Academy as it did for Thrun, but in Facebook, which showed how social interaction could live on the Internet.
"For courses where the goal is for the student to achieve mastery of a set of procedures (which is true of many courses in computer science and in mathematics), MOOCs almost certainly will change the face of higher education. Existing institutions that provide little more than basic, how-to instruction have a great deal to fear from MOOCs. They will have to adapt (and there is a clear way to do so) or go out of business."