Surprise Pythagorean Puzzle

This has been getting passed around fairly well, but in case you missed it, another wonderful enlightening problem from James Tanton (suitable and interesting for most grade levels):

Doing Mathematics (What It's Like)

I originally posted this last year when MAA first uploaded it to YouTube, but will revisit it now as a nice Sunday reflection via James Tanton, talking about the experience of doing mathematics:

James Tanton on Common Core

James Tanton has started a video series in support of Common Core. The first is below. Anyone involved in this debate should probably view it (important stuff!):

Epiphanies...

Love this newly-posted (by MAA) video of James Tanton answering the question, "What was the hardest thing you learned when studying math?" Especially timely to me since it ties in beautifully with the last two 'Sunday Reflections' I've posted here:

http://math-frolic.blogspot.com/2014/08/prime-synchrony.html
http://math-frolic.blogspot.com/2014/08/somethings-going-on-here.html

 

And, for more mathy stuff check out this Friday's link collection over at MathTango.

 

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."

Inspired by James Tanton

"Wild About Math" blog's latest entry in its "Inspired by Math" podcast series is a 46-min. interview with Princeton-educated, Aussie mathematician James Tanton. Hopefully you already know a bit about Dr. Tanton through his website and YouTube videos… but if you don't, you will surely want to learn more after listening to this podcast (at the end of the podcast-post is more link-info for Dr. Tanton, as well as information on his latest book, "Mathematics Galore!"):

http://wildaboutmath.com/2012/09/23/james-tanton-inspired-by-math-11/

Tanton's Principles

"5 key principles of brilliant mathematical thinking" from James Tanton:

http://www.jamestanton.com/?p=1097

It's a 4-part YouTube presentation so set aside some time to watch (under an hr.), but the 5 principles are essentially as follows:

1. visualize
2. employ common sense
3. intellectual curiosity and play
4. understand, don't just memorize
5. know... what you don't know

Pi = 2

A well-circulated paradox with a circle inscribed in a square results in π being equal to 4, but what I hadn't previously noticed was this similar James Tanton offering making pi equal to 2 (obviously considering these two paradoxes, pi must, on average, be equal to 3 ;-))

Latest From Equalis Community Blog

"Wild About Math" has a new blog entry up at the Equalis Community blog here:

http://www.equalis.com/members/blog_view.asp?id=565749

Always includes some interesting stuff... and in this instance, introduced me to another blog I was unaware of, "Grey Matters," which I've added to the blogroll at right. Also, blogger Sol mentions he'll be doing reviews of several of James Tanton's books in the future; something to look forward to. There is a link to the latest "Math Teachers At Play" carnival, and other amusements. Good stuff for a long weekend....

Some Miscellany For Wednesday

Another look at the foundations of mathematics in upcoming book:

http://tinyurl.com/33ylyvp

If you can't get enough of Pi:

http://facts.randomhistory.com/2009/07/03_pi.html

James Tanton's newly-active Twitter feed here:

http://twitter.com/#!/jamestanton

Dr. James Tanton

Kudos again to Sol, this time for introducing me to Dr. James Tanton, a creative mathematician with his own YouTube channel of interesting videos here (okay, I'm a sucker for an Aussie accent):

http://www.youtube.com/user/DrJamesTanton

His homepage website is here:

http://www.jamestanton.com/

...and "Math Mama" reviewed some of his work here:

http://mathmamawrites.blogspot.com/search?q=%22james+tanton%22

In other news, another recent "tweet" (from Twitter) that caught my eye:

 "It is known that e is irrational and that pi is irrational, but it is not known if their sum is irrational."

I'm wondering (maybe someone out there knows the answer) is it EVER the case that 2 irrational, transcendental numbers are known to sum to a rational???