Prime Progressions… (by 544,680,710)

Following up on yesterday's words from Richard Elwes I'll toss out another bit from his wonderful volume "Math In 100 Key Breakthroughs" (pg. 386). We're all familiar with "twin primes" like 11 & 13, and even triplet primes like 3, 5, 7. One can also signify longer sequences of primes that are separated by equal gaps: 11, 17, 23, 29, for example have spacing 6-apart (there are other primes, 13, 19, interspersed, but we're ignoring them).

Richard Elwes asks, "How long can sequences like this be?" and replies further, "The search quickly becomes hard, as the individual numbers involved become very large, too. The longest currently known arithmetic progression of primes consists of 26, beginning with 43,142,746,595,714,191 and then increasing in steps of 544,680,710. It has long been conjectured that there should be arithmetic progressions of primes of every possible length. This idea dates back at least to 1770, to the work of Edward Waring and Joseph Louis LaGrange. But the conjecture resisted all attempts at proof until 2004, when Ben Green and Terence Tao collaborated to prove their stunning theorem.
"If you want a list of 100 primes, each exactly the same distance from the last, the Green-Tao theorem guarantees there will be such a list somewhere. It does not, however, provide much useful information about where to start looking!"

...mind… blown… yet… again. . . .

…and, as long as we're speaking about primes, I hope most of you saw Web cartoonist xkcd's recent effort on the Goldbach conjecture(s):  http://xkcd.com/1310/

Finally, a safe, happy... and mathy NEW YEAR to one-and-all!!!

Meditating... on Chaos

Food for thought… an interesting passage from chapter 85 (on chaos theory, also known as "bifurcation theory") of Richard Elwes' "Math In 100 Key Breakthroughs" (pgs. 345-7):

"How can one produce a random number? In the late 1940's, John von Neumann proposed a very strange answer to that question.  He suggested that applying a simple algebraic rule a few times should do the job. The rule is to begin with some number, call it x, and then multiply x by (1 - x), and multiply the result by 4. That is to say: x --> 4 X x X (1 - x).
"There does not seem to be anything especially 'random' about this bit of algebra. Once the initial number is chosen, say x = 0.1, the result of applying the rule is then completely predetermined. But a little experimentation reveals von Neumann's insight. The sequence produced by this rule runs: 0.1, 0.36, 0.9216, 0.2890, 0.8219, 0.5854, 0.9708, and so on (each number given to 4 decimal places). There does not seem to be much of a pattern here, and in fact that is no illusion. You can extend the sequence for as long as you like and in fact no pattern will emerge. Someone who did not know the rule being used would find it virtually impossible to distinguish between this sequence and one produced by a genuinely random physical process such as radioactive decay."…

"Today, von Neumann's rule is known as the logistic map, and it is one of the simplest examples of mathematical chaos, a phenomenon which has been recognized in many different situations…."

"In von Neumann's pseudorandom number generator, everything rests on the number 4, known as the parameter. Changing that value completely alters the behavior of the system. If one replaces 4 with a new parameter of 2, the logistic map ceases to be chaotic. Instead, for any starting value, the sequence will quickly home in on a fixed value of 0.5. This is known as an attracting point of the system.
"Increase the parameter from 2 to 3.4, and something new occurs. After a while, the sequence will endlessly flicker back and forth between two values around 0.84 and 0.45. This is known as an attracting 2-cycle. Raise the parameter a little higher to 3.5, and this is replaced with an attracting 4-cycle, and then at 3.55, an attracting 8-cycle, and so on. As the parameter increases, the length of the attracting cycle keeps doubling 15, 32, 64, and so on. This behavior is what chaos theorists call a sequence of bifurcations."

He goes on to explain that the bifurcations end once the parameter hits a certain threshold value known as the Feigenbaum point (named after chaos theorist Mitchell Feigenbaum). Beyond that point (like "4" in the example) the produced sequence will act chaotically forever, producing the famous "butterfly effect" whereby two sequences beginning at only slightly different starting values "end up entirely unrecognizable from each other."

A Big Serving of Monday Potpourri

Catching up on a few things:

1) First, Keith Devlin continues his fascinating series on his MOOC experience (now 6 recent posts to catch up on in his latest series, if you haven't been following along):

http://mooctalk.org/

2) And if you missed Keith delightfully talking on NPR this weekend about the excitement generated by Yitang Zhang's attack on the Twin-prime conjecture earlier this year, give that a listen here:

http://www.npr.org/2013/12/21/256003488/and-the-number-of-the-year-is-the-lowly-2?ft=1&f=1007

3) Meanwhile, Peter Woit writes about the incredible difficulty involved in verifying last year's proof from Shinichi Mochizuki of the ABC conjecture (perhaps unparalleled in the history of math):

http://www.math.columbia.edu/~woit/wordpress/?p=6514

4) And if anyone has missed it, my own latest interview with popular mathematician James Tanton is currently up at MathTango:

http://mathtango.blogspot.com/2013/12/james-tanton-making-math-accessible.html

5) Perhaps mathematicians are too shy or uncomfortable talking about their (or their colleagues') politics, but I'm still interested to hear from you, if you aren't:

http://math-frolic.blogspot.com/2013/12/are-mathematicians-liberals.html

6) And three for your mathematical entertainment:

a) Hat tip to The Aperiodical for pointing out this fun 30-min. BBC podcast on math and magic:

http://www.bbc.co.uk/programmes/b03ls7y2

b) Just this morning, Presh Talwalkar posted a nice triangle geometry problem with both a traditional algebraic solution and a 'quickie' simpler solution available:

http://tinyurl.com/mtovhpx

c) And finally, a link that got some play on Twitter last week is this old "urban legends" of math discussion from mathoverflow.com:

http://tinyurl.com/l6gapx6
(some interesting, fun, and quite technical 'urban legends' included...)

Are Mathematicians Liberals?

A speculative posting today… just something I'm wondering about:

I follow a LOT of professional (PhD.) mathematicians around the Web, and from their occasional dabbling in political/cultural matters, my strong impression is that the vast majority could be categorized as political "liberals." One of our icons, Martin Gardner, in his recent autobiography, unabashedly labeled himself a "democratic socialist" and cites Norman Thomas as one of his "heroes."
At first I thought this made simple sense (I mean after all, aren't all astute, thinking individuals, liberals ;-))) but then I began to wonder… most (though not all) of the mathematicians I follow are 'academic' mathematicians -- they have a working association with some academic institution. Perhaps it is the academic milieu that makes one liberal, moreso than the field of mathematics???
There certainly are 'professional' PhD. mathematicians who work for private industry; so I'm curious what their political leanings are, and if they differ much from the academic crowd. Any thoughts…?

If anyone cares to respond to any of the following questions, I'd be interested just out of curiosity (and of course you can be 'anonymous'):

1) First, does anyone disagree that the majority of academic mathematicians could be characterized as political liberals?

2) If you agree with that characterization, is there anything inherent to the advanced study of math that encourages 'liberalism' (or is it the result of completely separate factors)? i.e., does 'mathematical thinking' tend somehow to promote liberal thinking?

3) If YOU are a PhD. mathematician working in private industry (or know of some) do you find any significant differences in your political views and those of your academic colleagues? (and if so, any speculation on why that is?)

These are pretty wide-open questions and generalizations, so I don't expect precise, rigorous answers.
Also, I know there have been studies or surveys done of political attitudes broken down by professions; just don't recall if any have ever specifically included "mathematicians" as a category -- if anyone is aware of such a survey, available on the Web, let me know.

Hmmm… Probability Puzzles

 I just recently discovered this blog which appears completely devoted to probability puzzles... making me feel just a tad like Homer Simpson… IF you substitute the phrase "probability puzzles" for "beer" ;-) :

Yin and Yang… No, Zipf and Zhang

A couple of pieces to catch up on if you've missed them:

Yitang Zhang, the largely unknown, humble New Hampshire mathematician who sort of broke open the twin prime conjecture earlier this year, is now deservedly being awarded the Frank Nelson Cole Prize in Number Theory come January (he has already received the Ostrowski Prize). Read another lovely news story about this unforeseen math champion here:

http://www.unionleader.com/article/20131215/NEWS04/131219488

And the below article notes that Zipf's Law is finding more applications to human society than its original linguistic roots with which I was familiar (however, note that several commenters take issue with the article's author):

http://tinyurl.com/lakytzh

The Value of Failure

In his latest MOOC blog posts Keith Devlin insightfully discusses expectations, real math, mathematical thinking, AND, the 'power of failure.'
According to Keith high school math and college math are "in many ways completely distinct subjects," and the "mathematical thinking" needed at the college level cannot be "taught" but must be "learned," which includes "learning by failing":

"...it is only when we fail that we actually learn something. The more we fail, the better we learn; the more often we fail, the faster we learn. A person who tries to avoid failure will neither learn nor succeed."

Read his post (part 2) if you need clarification on all that -- heck read both posts (much food for thought) even if you need no clarification! -- at the risk of sounding like a broken record, these ought not be missed if you're an educator:

http://mooctalk.org/2013/12/14/maththink-mooc-v4-part-1/

http://mooctalk.org/2013/12/15/maththink-mooc-v4-part-2/

ADDENDUM (12/17): now part 3 of this series from Keith is up: 
http://mooctalk.org/2013/12/17/maththink-mooc-v4-part-3/

By the way, elsewhere, Dr. Devlin cites J.K. Rowling's 2008 commencement speech at Harvard (having to do with "failure") as possibly the best commencement address ever. If you've never seen it, enjoy:

Move Over, Clifford Pickover

Richard Elwes has a new volume out, "Math In 100 Key Breakthroughs," that I'd add to the Holiday math book shopping list I've already posted. It's a bit reminiscent of Cliff Pickover's "The Math Book" -- I like a lot of Pickover's stuff, and he was kind enough to do an interview for me here, but I was never greatly enamored of that particular volume from Cliff, despite its wide success and popularity -- I do however like Elwes' effort to combine math text and gorgeous graphics in a delicious way, that flows along nicely.

Elwes' book runs essentially in chronological order and while the first third didn't grab my interest that much, covering earlier math history, it gets more interesting with coverage of more modern mathematics (say starting with Newton onward). The text is again (like Cliff's book) on the pithy side, but a bit more substantive than the latter; and I always find Elwes to be one of the very best, clearest, current explicators of mathematical ideas for a lay audience… all the more reason I wish he had gone just a tad more deeply into some of the subjects addressed here.
Still, the volume represents, I think, a splendid introduction to the variety and range of mathematics, especially for a young person with such inclinations. It is already 400 pages long (perhaps at least 1/3rd of that from graphics/pictures), so maybe further, pedagogic text would've added too much. While organized into 100 chapters or "breakthroughs," each chapter covers multiple specific topics, so there's a lot more than 100 topics touched upon here, and more, I think, than is covered in Pickover's choppy volume of 250 "milestones."

My one beef with the book is that there is no bibliography included (Cliff's book has one at the end)… or even better yet, would have been a "for further study" listing following each chapter, referencing sources to further the reader's interest/knowledge if so inclined. One might argue that because anyone can Google any subject these days and find copious additional material, such bibliographic references are no longer needed… but it is exactly because Google returns such copious, ill-prioritized suggestions, that a honed list of excellent selections from the author would be valuable.

Anyway, I highly recommend this beautiful book, especially if you liked Pickover's more coffee-table-like version… OR, even moreso if you didn't find Pickover's volume satisfying, but still fancy the concept of combining wide-ranging, informative mathematical text with beautiful illustrations.

A > B > C > A

In time for the new year perhaps.…

You may be able to order a set of James Grime "nontransitive dice" (earlier sold out, but being re-supplied):

http://mathsgear.co.uk/collections/all-products/products/non-transitive-grime-dice

Nontransitive dice (also known as "Efron Dice" after one of the inventors) have probably regained some attention since being mentioned in Simon Singh's recent book, "The Simpsons and Their Mathematical Secrets." Several different nontransitive combinations are actually possible, but the set mentioned in Singh's book, include "Die A" with sides, 3,3,5,5,7,7, "Die B" with sides 2,2,4,4,9,9, and "Die C" composed of sides 1,1,6,6,8,8.  On average, a throw of Die A will beat (56% of the time) a throw of Die B, and a throw of Die B will beat (56% of the time) Die C… YET, Die C, on average, will beat out Die A (56% of the time)… How cool is THAT! or as, Singh writes, "Nontransitive relationships are absurd and defy common sense, which is probably why they fascinate mathematicians."

Wikipedia covers the subject here:

http://en.wikipedia.org/wiki/Nontransitive_dice

But of course James Grime is far more entertaining here:

http://singingbanana.com/dice/article.htm

Fans of the simple competitive game "Rock, paper, scissors," may recognize that it too operates on the basis of nontransitivity.


On a sidenote, the current post over at MathTango contains some puzzle fun, including my re-tell of an old favorite from Raymond Smullyan:

http://mathtango.blogspot.com/2013/12/puzzles-puzzles.html

AND finally, a reminder that Sunday will be the last day of the sparsely-entered :-( caption contest over at MathTango:

http://mathtango.blogspot.com/2013/11/caption-contest.html

Why oh Why?

"Why Write a Math Blog" -- that was the title of a recent short post at Ken Abbott's blog, where he offered 3 reasons for his blogging:

http://www.math-math.com/2013/11/why-write-math-blog.html

His thoughts inspired me to think about my own reasons for math-blogging, since my sparse (academic) math background makes it an even more interesting question for me… and my reasons overlap, but differ a bit, from Ken's.
Indeed, when friends have asked, in a surprised tone, "why" do I math blog my first reaction is to explain that while I write ABOUT math, I don't actually DO much mathematics on the blog. I'm more interested in the topic of math and mathematicians than in the working out of math, for which I have limited competency.

So reasons for math-blogging here at Math-Frolic are:

1)  This blog was born with the demise of Martin Gardner back in June 2010 -- I was doing a small science blog at the time, but Gardner's death brought back memories of the hours of enjoyment I got from his work, and I decided to begin anew with a blog focusing on math in his honor. (expecting it to be short-lived, but over time it grew, as to my naive amazement there seemed to be never-ending material to draw from.)

2)  I like trying to make math interesting to others.

3)  I enjoy curating information, and writing.

4)  In the process of doing the blog I get to learn a great deal myself.

5) And best of all, mathematicians are among the most interesting people in the world to me, and the blog puts me in contact with people I would otherwise never have had the pleasure of crossing paths with!

So HOORAY to the world of math-blogging! (I'm sure other math-bloggers have their own reasons for blogging, sometimes different from the above -- feel free to chime in with your own motivations/rewards in the comments).

Excitement... from 1984

The math bibliophile in me is always excited when our local public library holds a used book sale... and today my lucky find was "Mathematics: People Problems Results," a 3-volume anthology set from 1984, edited by Douglas Campbell and John Higgins. It looks to be a scrumptious set of ~90 rich essays (including some classics) from a great panoply of superb (and famous) writers/mathematicians. As I've said before, math is so timeless that even a 30-year-old book-set like this can contain fabulous stuff.

This all reminds me that the current caption contest over at MathTango only has two entrants thus far and I'll probably end it next Sunday (the 15th), so give those folks some competition and get your entries in.

Singh, Simpsons, Science Friday

If you missed Simon Singh (author of "The Simpsons and Their Mathematical Secrets") and David Cohen on NPR's "Science Friday" today with Ira Flatow, well, you're in for a treat:

http://tinyurl.com/kdys4wz

Weird Math...

What do the two numbers, 70 and

26,963,672,211,957,831,828,322,834,071,143,299,817,754, 720,290,127,404,079,937,026,385,368,922,075,196,690,720,690,562,498,337,038,657,263, 353,255,952,256,005,850,803,053,091,152,216,128,172,198,270,512,414,580,092,743,322, 379,544,478,286,025,897,899,890,351,444,085,611,625,835,160,270,418,964,124,507,243, 890,975,821,522,176,465,361,680,177,670,297,930,314,037,850,339,675,559,057,554,452, 347,547,946,165,134,639,879,111,112,583,151,946,671,967,876,920,506,598,818,088,728, 910,330,021,016,856,674,391,763,268,224,262,067,132,913,691,721,407,174,127,885,521, 288,146,239,271,038,154,486,086,650,600,357,888  ...have in common?

They are both "weird" numbers… and I mean that, in a technical way! 70 is the smallest "weird" number and that second monstrosity is, to date, the largest known (of an infinite number) of weird numbers, at 226 digits. It was found by these Central Washington University folks:

http://www.kimatv.com/news/local/CWU-math-students--234496131.html
also see here: http://tinyurl.com/knzfswc

"Weird" numbers are those natural numbers whose divisors add up to more than the number itself, and for which NO selection of divisors sum exactly to the original number [for example, for 70, the divisors are 1, 2, 5, 7, 10, 14, and 35, which sum to 74, and no possible combination adds exactly to 70]. The student group originally discovered the first new weird number in over three decades, with a 72-digit find, before eventually reaching the above record.  Per the article, "a better understanding of weird numbers leads to a better understanding of factorization, which is the basis of all modern cryptography." [in case you were wondering of what possible use this could be!]

Here is a sequence of weird numbers from the OEIS directory:

70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17990, 18410, 18830, 18970, 19390, 19670

Interesting that all of these, with the single exception of 836, end with a "2" or a zero, yet the new record find ends with an 8. -- I have no idea what the distribution of end-digits is for the full panoply of currently-known weird numbers??? (It is also not known with certainty if ANY odd weird numbers exist... but if they do, they must be very, VERY large!)

[I don't know if it's even possible to explain at a layperson level, but if someone in-the-know wants to try and explain in the comments what sort of method/algorithm one employs to discover weird numbers of such length (or alternatively how one verifies such a number) I'd be curious to hear it.]

But Tell Us How You Really Feel, Dr. Z

Communicating math… aye, there's the rub:

Rutgers professor Doron Zeilberger has a bit of a gadfly/curmudgeon reputation in the mathematical community… which is what (in part) makes him so interesting to hear out! He has an opinion piece about math communication in the latest "Notices of the AMS" which is getting some buzz, including inspiring a blog post from Jason Rosenhouse. The Zeilberger letter (pdf) is here:

http://www.ams.org/notices/201311/rnoti-p1431.pdf

In it, he criticizes "pure mathematics" for its 'fanatical' focus on "rigorous proofs," and urges greater emphasis on "experimental mathematics" noting:

"Mathematics is so useful because physical scientists and engineers have the good sense to largely ignore the 'religious' fanaticism of professional mathematicians and
their insistence on so-called rigor, which in many cases is misplaced and hypocritical, since it is based on 'axioms' that are completely fictional, i.e., those that involve the
so-called infinity.
"The purpose of mathematical research should be the increase of mathematical knowledge, broadly defined. We should not be tied up with the antiquated notions of
alleged 'rigor'."

It is an interesting (recommended) read and not very long, and his criticism of higher-level math communication as "highly dysfunctional" (only comprehensible to the few specialists who share a given area of work), spurred Jason Rosenhouse to write his own interesting blog post, largely in agreement:

http://scienceblogs.com/evolutionblog/2013/12/03/communicating-mathematics/

Rosenhouse's piece would be worthwhile alone for his own illustrative account of a math conference presentation as a nervous grad student; but there are many other good points in the post as well.

I recommend you read both these pieces, and you may want to also check out some of Dr. Zeilberger's other (wishy-washy, NOT!)  opinions here:

http://www.math.rutgers.edu/~zeilberg/OPINIONS.html

The Oracle of Devlin ;-)

I wish that just once Keith Devlin would write a blog post that I could yawn at and didn't feel obligated to refer my readers to. But the man just seems incapable of writing anything mundane or trite or ordinary. His latest thoughtful offering, on MOOCs and "quantitative reasoning," here:

http://devlinsangle.blogspot.com/2013/12/moqr-anyone-learning-by-evaluating.html

I love watching Dr. Devlin's experience with MOOC-building evolve over time, and his openness/honesty in letting us observe as he rides the roller-coaster of hope/doubt/optimism/pessimism/confidence/uncertainty that seem to coincide with the development of MOOCs (if not education change/reform in general!!)
He will be substituting something he calls "Test Flight" in place of a final exam in the next iteration (beginning Feb. 3) of his own mathematical-thinking MOOC, and watching to see if it succeeds or 'crashes and burns.'

He winds down this particular piece with these contemplative words:

"The more people learn to view failure as an essential constituent of good learning, the better life will become for all. As a world society, we need to relearn that innate childhood willingness to try and to fail. A society that does not celebrate the many individual and local failures that are an inevitable consequence of trying something new, is one destined to fail globally in the long term."

A Stocking Stuffer, Perhaps

On impulse about a month ago (and trying to use up a discount-coupon ;-), I purchased a little math reference volume at my local Barnes and Noble, entitled "Math In Minutes: 200 Key Concepts Explained in an Instant" by Paul Glendinning (or "Maths In Minutes" for the British version). Turns out it was published in 2012, even though I didn't see it 'til a few weeks back (as a British offering it may have taken awhile to reach the States).
Anyway, wasn't planning to mention it here, but occurs to me it might make an okay stocking stuffer for some budding math person on your holiday shopping list so I'll give it a plug. At about 5"x 5"x 1" it will literally fit in some oversized fireplace stockings! Amazon describes it, in part, thusly:

"...simple and accessible... introduction to 200 key mathematical ideas... described by means of an easy-to-understand picture and a maximum 200-word explanation… Compact and portable format -- the ideal, handy reference."

"Ideal" is probably too strong a word, but definitely "handy." The format actually is reminiscent (in miniature) of Clifford Pickover's wildly-popular "The Math Book," in so much as there is generally a brief text on the left-hand page followed by a pertinent (black-and-white) picture on the right-hand page. Not as glossy or beautiful as Cliff's work, and the text is even thinner (indeed, rather superficial) than Pickover's, but the trade-off is a very portable, bite-size volume, that still touches a lot of ground. Also, unlike Pickover's strictly chronological format, the Glendinning offering categorizes its 200 ideas into broader topic areas:

- Numbers
- Sets
- Sequences and series
- Geometry
- Algebra
- Functions and calculus
- Vectors and matrices
- Abstract algebra
- Complex numbers
- Combinatorics
- Spaces and topology
- Logic and proof
- Number theory

Again, it's not a great book you need to rush out and get for your own shelf, but perhaps a fun little reference/gift for some young mathematician you want to surprise.

A Few Quotes

Just a few quotations for your reflection today, taken from Simon Singh's "The Simpsons and Their Mathematical Secrets":

From famous British mathematician G.H. Hardy: "Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. 'Immortality' may be a silly  word, but probably a mathematician has the best chance of whatever it may mean."

From Simpsons writer (and Harvard physics grad) David Cohen, on the satisfaction derived from slipping mathematics into Simpsons' episodes: "I feel great about it. It's very easy working in television to not  feel good about what you do on the grounds that you're causing the collapse of society. So, when we get the opportunity to raise the level of discussion -- particularly to glorify mathematics -- it cancels out those days when I've been writing those bodily function jokes."

And from author Singh: "It would be easy for non-nerds to dismiss the mathematical shenanigans that appear on The Simpsons and Futurama as superficial and frivolous, but that would be an insult to the wit and dedication of the two most mathematically gifted writing teams in the history of television. They have never shied away from championing everything from Fermat's last theorem to their very own Futurama theorem.
"As a society we rightly adore our great musicians and novelists, yet we seldom hear any mention of the humble mathematician. It is clear that mathematics is not considered part of our culture. Instead, mathematics is generally feared and mathematicians are often mocked. Despite this, the writers of The Simpsons and Futurama have been smuggling complex mathematical ideas onto prime-time television for almost a quarter of a century."

"Futility Closet"... the BOOK!

A couple posts back I mentioned some books worthy of your consideration during the Holiday shopping season... you can now add another: turns out Greg Ross has put together a compendium of his best miscellaneous, quirky "amusements" for a "Futility Closet" book (this is GREAT news!):

http://www.amazon.com/dp/0989887103

Here's a nice, recent geometry tidbit from the Futility Closet blog:

http://www.futilitycloset.com/2013/11/25/some-odd-theorems/

And for those who missed it, I interviewed Greg at Math-Frolic just about one year ago:

http://math-frolic.blogspot.com/2012/11/greg-ross-of-futility-closet.html

Also, reminder, only couple of entries thus far to win some books over at my MathTango caption contest:

http://mathtango.blogspot.com/2013/11/caption-contest.html

Something for the Young, the Older, and the Conspiracy Theorist

A few items you may have already seen, but if not...:


1) Sherlock-Holmes-wannabes are still trying to figure out who "Satoshi Nakamoto," creator of Bitcoin really is. The NY Times piece below draws an unconvincing (I think) link to the former proprietor of the Silk Road website. The guessing game may be more fun than learning the true identity will ever be.

http://tinyurl.com/otavjw3

2) Another great post from MathMunch covering several matters, but what most interested me (because I'd not heard of it) is a Scrabble-like game called "Numenko" that helps young people learn/practice arithmetic:

http://mathmunch.org/2013/11/04/numenko-turning-square-and-toilet-paper/

3) And finally, my favorite recent read from the Web, the always excellent Natalie Wolchover attempts to explain (better than I've ever seen done before, though still a tough-read) two different approaches to solving the infinity "continuum" problem:

https://www.simonsfoundation.org/quanta/20131126-to-settle-infinity-question-a-new-law-of-logic/

Set aside some brain-time to take in this account of "forcing axioms" versus "V = ultimate L".

WHAT A YEAR!

…in popular math writing!! One thing I'm grateful for this Thanksgiving week is all the, not merely good, but GREAT math reads that appeared for the public this year; many of which I would put into "best of" categories:

For my money...:

BEST math-related book seen in decades: "The Outer Limits of Reason" by Noson Yanofsky
MOST fun-and-entertaining book in a long time: "The Simpsons and Their Mathematical Secrets" by Simon Singh
RICHEST introduction to high level mathematical ideas in a general audience book: "Love and Math" by Edward Frankel
MOST highly anticipated math biography evuh!: "Undiluted Hocus-Pocus" by Martin Gardner
BEST popular math anthology for a general reader: "The New York Times Book of Mathematics" edited by Gina Kolata
BEST (though hard to choose) book from the prolific Ian Stewart: "Visions of Infinity"
Probably my favorite popular treatment of statistics for a mass audience: "Naked Statistics" by Charles Wheelan
One of the best books ever focused entirely on a Clay Institute Millennium Problem: "The Golden Ticket" by Lance Fortnow (...this is a tough call though since there were several Millennium Problem books this year)

(I've reviewed all the above volumes over at MathTango at some point, with the lone exception of Singh's book, which is an absolute blast to read, but received SO MUCH positive coverage/publicity I feel no need to do a review myself.)

If there's a math geek on your Holiday shopping list I don't think you can go wrong with some of the above picks. It's amazing how a subject so often perceived as dry-and-dull as math, continues generating so many great volumes, demonstrating what a living, growing, kick-ass subject it really is ;-)… hats-off to the authors and publishers who keep the books flowing!!!

Please add your own favorites from the year-now-ending in the comments below. And feel free to add 2013 volumes of a more technical nature as well, if you think math professionals/specialists should be made aware of them.

...Meanwhile, as an aside, check out MathTango for the Caption Contest that is now up there!:

http://mathtango.blogspot.com/2013/11/caption-contest.html

Nature, the Ultimate IT Manager

Nice weekend read....

http://nautil.us/issue/7/waste/nature-the-it-wizard

Somewhat more to do with biology than mathematics, but the above article from Nautilus, on 'Nature as an IT wizard,' is simply too rich not to pass along. It crosses into information-processing, neuroscience, and even physics, with plenty of mathematical implications. Subtitle for the piece is, "Nature manages information, the currency of life, with exquisite efficiency." Three quick bits:

"Every organism is a brief upwelling of structure from chaos, a self-assembled wonder that must jealously defend its order until the day it dies. Sophisticated information processing is necessary to preserve and pass down the rules for maintaining this order, yet life is built out of the messiest materials: tumbling chemicals, soft cells, and tangled polymers."

"Not only does DNA store information at a density per unit volume exceeding any other known medium, it can achieve one quarter of the maximum information density allowed by the laws of physics (set by the entropy of a black hole). It’s so dense that all the world’s digital data could be stored in a dot of DNA the weight of eight paper clips. This remarkable storage density is paired with an equally remarkable reading mechanism."

"Why would nature use fractal geometry so regularly? Mathematically, fractals are interpreted as having a fractional dimension higher than the space they reside in: A fractal drawn on a two-dimensional sheet of paper, for example, has a higher dimension—say, 2.1. This is a useful feature, allowing nature to pack some part of a fourth dimension into three-dimensional space."

A Little Bit of Statistics Is a Dangerous Thing

Well, I've had a few jabs at statistics this week in posts, so may as well end with another, more strictly humorous one. This is a joke I'd not seen before that comes verbatim from Simon Singh's wonderfully entertaining book "The Simpsons and Their Mathematical Secrets":

While heading to a conference on board a train, three statisticians meet three biologists. The biologists complain about the cost of the train fare, but the statisticians reveal a cost-saving trick. As soon as they hear the inspector's voice, the statisticians squeeze into the toilet. The inspector knocks on the toilet door, and shouts: "Tickets, please!" The statisticians pass a single ticket under the door, and the inspector stamps it and returns it. The biologists are impressed. Two days later, on the return train, the biologists showed the statisticians that they have bought only one ticket, but the statisticians reply: "Well, we have no ticket at all." Before they can ask any questions, the inspector's voice is heard in the distance. This time the biologists bundle into the toilet. One of the statisticians secretly follows them, knocks on the toilet door and asks: "Tickets please!" The biologists slip the ticket under the door. The statistician takes the ticket, dashes into another toilet with his colleagues, and waits for the real inspector. The moral of the story is simple: "Don't use a statistical technique that you don't understand."

On a related note I just recently discovered this webpage which focuses on "Simpsons math" as well:

http://mathsci2.appstate.edu/~sjg/simpsonsmath/

Of Primes and Probability

Two bits for a Wednesday...:

1) In a fascinating long-read for Quanta Magazine, Erica Klarreich covers the astounding progress so far made in the "prime number gap" of the Twin Primes Conjecture, in just six months since Yitang Zhang postulated a limit to the gap of 70 million!:

http://tinyurl.com/pw6hyyz

James Maynard, a post-doc, has wrestled the gap down to no more than 600, well below the result that even Terry Tao's Polymath group had yet achieved.

an excerpt to whet your appetite:

"Zhang’s work and, to a lesser degree, Maynard’s fits the archetype of the solitary mathematical genius, working for years in the proverbial garret until he is ready to dazzle the world with a great discovery. The Polymath project couldn’t be more different — fast and furious, massively collaborative, fueled by the instant gratification of setting a new world record.
"For Zhang, working alone and nearly obsessively on a single hard problem brought a huge payoff. Would he recommend that approach to other mathematicians? 'It’s hard to say,' he said. 'I choose my own way, but it’s only my way.'
"Tao actively discourages young mathematicians from heading down such a path, which he has called  'a particularly dangerous occupational hazard' that has seldom worked well, except for established mathematicians with a secure career and a proven track record. However, he said in an interview, the solitary and collaborative approaches each have something to offer mathematics.
“ 'It’s important to have people who are willing to work in isolation and buck the conventional wisdom,' Tao said. Polymath, by contrast, is 'entirely groupthink.' Not every math problem would lend itself to such collaboration, but this one did."

2) I mentioned a couple of posts back that in the Preface to his new book ("Will You Be Alive 10 Years From Now?") Paul Nahin gives an example of a Marilyn vos Savant column where the famous Mensa-ite gives the WRONG answer to a math question and sometime later corrects herself. The question, and her initial ill-fated answer ran as follows:

Q.: "I manage a drug-testing program for an organization with 400 employees. Every three months, a random-number generator selects 100 names for testing. Afterward, these names go back into the selection pool. Obviously, the probability of an employee being chosen in one quarter is 25 percent. But what’s the likelihood of being chosen over the course of a year?"

A.: "The probability remains 25 percent, despite the repeated testing. One might think that as the number of tests grows, the likelihood of being chosen increases, but as long as the size of the pool remains the same, so does the probability. Goes against your intuition, doesn’t it?"

Nahin points out that the actual probability of being chosen at some point during the four quarters of testing works out to 0.6836, considerably greater than Marilyn's 0.25.

In a later column Marilyn 'fessed up, "My neurons must have been napping" and corrected herself:

http://www.parade.com/45916/marilynvossavant/22-sunday-column-2/

You can also see the math involved at this Forum site where the problem was discussed:

http://www.aiqus.com/questions/34413/ask-marilyn-new-probability-question-controversy

MOOCs as "Silicon Valley's Next Grand Challenge"

Professor Keith Devlin once again (this time in Huffington Post) with a thoughtful post on the future of MOOCs:

http://www.huffingtonpost.com/dr-keith-devlin/why-moocs-remain-silicon-_b_4289739.html

Wish I could just quote the whole thing, but I'll leave you with these bits:

"The fact is, Silicon Valley has yet to come to terms with education... A lot of what goes on in good (sic) education is almost certainly not scalable. That means the familiar hockeystick growth in users that can result in a hugely profitable IPO or buyout is not likely. On the other hand, as companies like Pearson and Apple know very well, the market for any particular educational product renews every twelve months as children and young adults move through the system."

"[MOOCs] are not 'regular university courses online' and they won't replace universities. They may well, however, reach a stage where they disrupt higher education, and if so, institutions that don't adapt to a changing landscape are indeed likely to go out of business."

"...there you have tomorrow's talent supply. Those huge [MOOC] dropout rates that were once regarded as a big problem turn out to have been our first glimpse of an amazing global filter for people with commitment, persistence and ability."

"MOOCs do not and, I believe, cannot replace a good university education. But they can, and in some cases already have, provide a pathway to such education for millions of people around the world who, for various reasons, do not at present have any access. Scale that across the entire world, and you have disruption."

...Meanwhile, for any interested, and who haven't already seen it, my overview of Noson Yanofsky's "The Outer Limits of Reason" (a fabulous volume I heartily recommend to all science-types) is now up at MathTango:

http://mathtango.blogspot.com/2013/11/phenomenal-book.html





 

Probability, Oy Vey....

The Preface to Paul Nahin's new book, "Will You Be Alive 10 Years From Now," includes several wonderful probability problems… I'll likely use them here over time, and will start with the one that seems the most problematic even though on the surface it appears simple.

Columnist Marilyn vos Savant is famous for introducing the 'Monty Hall problem' to the public and giving the correct answer even when many professional mathematicians were initially labeling her "wrong."

Nahin argues that in a different example she WAS wrong, though I think he simply misinterprets matters. The initial, simple question that a reader asked Marilyn in this case was:

“Say you plan to roll a die 20 times. Which result is more likely:
(a) 11111111111111111111 or (b) 66234441536125563152”

Marilyn answered that both sequences were equally likely as outcomes from such a procedure… there is little controversy over that answer (and Nahin agrees with it)… from a strictly frequentist view of rolling a fair die, all sequence-outcomes being equally likely (that likelihood being very small, BTW). BUT, then Marilyn went on to note, “But let’s say you rolled a die out of my view and then said the results were one of those series. Which is more likely? It’s (b) because the roll has already occurred. It was far more likely to have been that mix than a series of ones.”

I don't really have much difficulty with that answer either, but Nahin takes her to task claiming the answer is "wrong" and the probabilities are still equal… that "rolling the die out of view" has no consequence. But clearly there is a difference between anticipating in advance a resultant sequence out of ALL the possible sequences that a procedure might produce, versus addressing just two given sequences AFTER a procedure has already taken place. Nahin faults vos Savant for essentially changing the original question, BUT she clearly states that that is what she is doing (in order to make what I think is an interesting and worthwhile point; it's almost a sort of frequentist vs. Bayesian distinction).
[One way to think about it is simply to make the sequence more ridiculously long: suppose I roll a FAIR die a million times; I record the results and tell you that the outcome was either a million ones, OR, some more-random-looking list of figures… prior to rolling the die both sequences would be equally likely, but with the task already completed, and ONE of the TWO given choices GUARANTEED to be the actual sequence, the second one is more probable.]

Anyway, I find this a good example of how the semantics of a probability problem is often trickier than the math or logic involved (similarly, the precise way the Monty Hall problem is stated and understood is crucial in reaching the right answer under the variety of exact set-ups that can be proposed for it).

[p.s. --  Nahin's Preface offers a second example of vos Savant getting a problem wrong (and later correcting herself), and at some point I'll give that example as well.] [Now, HERE.]

(image credit: Personeoneste/WikimediaCommons)

ADDENDUM: Below are Marilyn's two original responses to this problem in Parade magazine:

http://www.parade.com/45638/marilynvossavant/sundays-column-07-31-11/
http://www.parade.com/45746/marilynvossavant/sundays-column-10-23-11/

Sigma Schmigma...

Not sure which is more entertaining: reading another rant against statistical significance methods in the press... or, seeing the economist-author of such get taken to the woodshed by physicists in the comments section:

http://opinion.financialpost.com/2013/06/10/junk-science-week-unsignificant-statistics/

Learning, Books, Puzzles, P-values

Trying to get my blogging energy back after the 104th Carnival of Math sapped more of it than I expected 8-/
Will just point to a few pieces I enjoyed over the last few days:

1) The always-interesting Jo Boaler writes in The Atlantic about how the controversial Common Core approach to math education could help break down "math stereotypes." She believes the Common Core curriculum can produce students with more math confidence "who can develop mathematical models and predictions, and who can justify, reason, communicate, and problem solve… who are powerful mathematical thinkers and who have not been held back by stereotypical thinking and teaching":

http://tinyurl.com/lnmyv7r
(...of course not everyone in the 250+ comments agrees with her ;-)

2) Yet another wonderful podcast interview with Edward Frenkel (author of "Love and Math") this time from The Guardian:
http://tinyurl.com/kqrp5tw

3) "I just don't get no respect anymore"… THAT's what statistical p-values must be saying to themselves these days... a couple more articles critical of their standard use:

http://tinyurl.com/pcmfo2w (from Ars Technica)
http://tinyurl.com/mm9aceb (from Scientific American)

4) And an interesting little probability puzzle from Presh Talwalkar here:

http://tinyurl.com/lnpveb7

5) Speaking of probability, Paul Nahin fans will be delighted to see that Dr. Nahin has a new book out, "Will You Be Alive 10 Years From Now?" -- frankly, I think it's a crummy title, but nonetheless looks like a fantastic/entertaining read if you enjoy probability puzzles (which is the entire focus of the volume -- I'll likely employ some of the content for blogpost material here eventually).
Don't know when I'll find time to review it though, since my current priority is Noson Yanofsky's "The Outer Limits of Reason" which I regard as the most important popular math-related volume in quite some time.

Come Frolic at The 104th Carnival of Math!

The 104th (...or, 1101000st if binary is your thing) Carnival of Monthly Mathiness has arrived for your perusal and delectation... this month's Carnival 'ballooned' to a chock-full 40 diverse entries!... so grab some cotton candy (or a cup-o-exquisite-fresh-perked-hand-warming-eye-opening-dark-roast-Kenyan-or-Costa-Rican-aromatic-java... with a dash of soymilk should you desire) and come set a spell:

104 is the sum of eight consecutive even numbers, count 'em eight!... 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20

...but enough technical chit-chat; it's Thanksgiving Holiday month for American readers so without further adieu we'll give gratitude to these posts for strutting their stuff:
(...and please let me know ASAP of any errors below or mis-working links.)
 

October was a month full of Martin Gardner remembrances (his birthday being Oct. 21, and his autobiography recently released). One example was "Celebrations of Mind Honor Math's Best Friend, Martin Gardner" from Colm Mulcahy for Scientific American blogs.
Gardner of course demonstrated that doing math could also mean having fun... some other examples:

'The boy born on Tuesday' problem is one of the most famous math puzzles since Monty Hall. Rob Eastaway gives his take on it in "The Irksome Tuesday Boy Problem."  [As can be found on Google, this problem has generated a LOT of discussion over the years.]

Speaking of puzzles, "On Time" is a 'clock' puzzle ("mind-reading trick") from the always wonderful Futility Closet.

In "Some Geometry Notes on a Babylonian Square Root" Pat Ballew of Pat's Blog speculates about the Babylonian use of geometry to solve an algebraic problem.
I'm also submitting Pat's post, "Great Problems For High School" which includes several classic problems.

In "Heroic Triangles" Colin Beveridge of Flying Colours Maths offers a nice little bit of triangle geometry. And then in "The Geometry of Sec" Colin elucidates some trigonometry for us, as well.
And finally, another Colin entry this month was "Why I Loved the MathsJam Conference," reporting on the awesome annual gathering of ~100 mathematicians-at-play at MathsJam in Britain.

In the post "From the Mailbag: Dual Inversal Numbers" Katie Steckles of The Aperiodical, reports on a budding 9-year-old mathematician taking note of "an interesting property of numbers." 

I found this Brian Hayes piece (from Bit-player), "The Keys to the Keydom," on a flaw in RSA encryption, quite interesting.
Also interesting to watch has been the rise/fall/rise/.... of Bitcoin currency, especially with some of its recent problems, and Ed Felten of Freedom To Tinker blog tells us that "Bitcoin Isn’t So Broken After All." 

In "Breakfast at Les Deux Magots" Matifutbol graphically explains the "friendship paradox" for those of you dwelling over why your friends have more friends than you do.

I've never understood the trendy interest in Zombies and Evelyn Lamb seems to find it curious as well, as she looks at one researcher's attempt to analyze that trend/obsession in:  "Zombie Fever: a Mathematician Studies a Pop Culture Epidemic."
 
And finally, and only for the most serious amongst you, Tim Gowers did a looong (and deep) October post regarding a poly-mathematical approach he was interested in trying on the P vs. NP problem: "What I Did In My Summer Holidays" (warning: you can probably read ALL the other posts in the Carnival in the time it may take you to read and digest Tim's post! -- p.s., I selected this (a great example of Tim's writing), Tim didn't submit it.
----------------------------------------------

Meanwhile, three Peter Rowlett posts were contributed this month:
"Council Orders Maths and Sudoku To Be Removed From Mathematician’s Gravestone" was a quirky news item that is almost self-explanatory. 
In "Emergency Maths Arcade…" Peter suggests math games you can enjoy when all you have available is pen and paper (...like, uhhh, back in the days when I grew up!).
And finally, "Recent History," touches upon some "recent"(?) results in mathematics.

Speaking of history, Thony C. sends in the Aldres Caicedo post, "Credit," from A Kind Of Library blog, concerning Johann Bernoulli's contributions to the early days of calculus.
And some more history sent along by Thony: a few days ago was the anniversary of philosopher Gottlob Frege's birth... a post from Yovisto blog briefly synopsizes the great logician's achievements: "Gottlob Frege and the Begriffschrift."

 In "Lost In the Fourth Dimension," Lee Randall manages to link together The Simpsons, Twilight Zone, and her own childhood, in a post from her blog, "A History of My Life In 100 Objects."

Evelyn Lamb submitted a couple of Kate S. Owens posts: In "Combinatorics and Pampers," Kate is troubled by the inefficiency of 15-digit product codes used on Pampers diapers. Perhaps Proctor-and-Gamble (maker of Pampers) knows something about the coming baby boom the rest of us don't know!?
And in "Mathematics in Fiction Class Visit" Kate touches upon both gender and semantic issues amidst a discussion of the difference between "mathematicians," "math teachers," and "math professors." 

Speaking of gender-related matters, Yen Duong (Baking and Math blog) relates her recent experience being the only female in a meeting of math academics in "Surprisingly Emotional Reaction to Being a Woman in Math." 

...and, on to sports: for NBA fans (and coaches!) Tallys Yunes' O.R. By the Beach blog offers a post attempting to maximize NBA winning percentages based upon the analysis of playing and resting time for players: "Optimally Resting NBA Players." 

And in another sports note, Laura McLay of Punk Rock Operations Research explains "Why the Bears Should Have Gone For It on Fourth and Inches." 

"In Love... With Math" was my own thumbs-up overview at MathTango of Edward Frenkel's well-publicized book, "Love and Math" which focuses on the "Langlands Program," while extolling the author's love for mathematics.

Education and learning math were, as always, frequent subjects of posts this month: 

Keith Devlin's latest Devlin's Angle post is a great piece, entitled "The Educational Power of Elementary Arithmetic" on the controversy surrounding American math education.


In "Math Munching Today," Beth Ferguson of Algebra's Friend tells of using MathMunch in her classroom (along with Edmodo) to get students working/thinking a bit beyond the usual secondary curriculum. 

From Ben Orlin's Math With Bad Drawings blog a lovely introduction for young people to statistics, called "The Bear In the Moonlight."

Sue VanHattum of Math Mama Writes tackles explaining calculus to students in a series of posts, including "What Is Calculus, Part One."

Aatish Bhatia's "The Math Trick Behind MP3's, JPEGs, and Homer Simpson's Face" at Nautilus, has been cited as one of the clearest, audience-friendly explanations of Fourier transforms to come along in awhile.

In "My Favorite Christmas Present" Kevin Knudson uses an old math book to bring logarithms into focus as an algebraic tool (despite the 'blank stares' they draw from his students!).

Over at MrHonner, in the post "Order These Things From Least to Greatest" Patrick Honner contemplates how the potential answers to a given problem (or rather, HOW to grade those answers) can be more interesting than the problem itself. 

In a tweet, Steven Strogatz called Gary Rubinstein's post, "The Death of Math,"  "The smartest piece I've ever read about math reform," which I figure earns it inclusion in the Carnival!

On a related topic, Philip Stark at The Berkeley Blog offered an analysis of teacher effectiveness in "What Exactly Do Student Evaluations Measure?" 

In "Snowballing Good Questions" Alex Overwijk of SlamDunkMath gives his class a lesson in how to develop good questions.
----------------------------------------------

A few visuals (not blog posts) were also submitted this month, adding still further variety:

First, a video entitled "A Math Major Talks About Fear" from Saramoira Shields -- it was retweeted by so many folks I felt it deserved a spot in the Carnival.
Similarly, Kit Kilgour suggested another very popular video this month that appeared on Vimeo, "Beauty and Mathematics" -- the beauty and ubiquity of math captured in less than 2 minutes.
Meanwhile, Katie Steckles suggested 2 other videos for your attention:
"Prism Marching Band" a short imgur piece you need to see to appreciate.
...and from YouTube, "Sorting Algorithms" which lo-and-behold, involves, sorting algorithms.


 And, on a different note, I'll close out linking to the latest interview in my own series at MathTango, with computer scientist Dr. Noson Yanofsky, author of "The Outer Limits of Reason," a fantastic, multi-subject book Shecky highly recommends to ALL!!


Wow!... Hope EVERYone finds at least a few things addressing their interests in this month's wide-ranging Carnival! And THANKS! to all who contributed....

(note: ...I've included all posts that I received as submissions, so if by any chance you sent in an entry and it doesn't appear here, then it got lost in the shuffle along the way... please let me know so I can insert it.)
----------------------------------------------

==> If you need to catch up on your Carnival reading, the prior (103rd) edition is at Evelyn Lamb's Roots of Unity HERE, and the next Carnival will be hosted by Oluwasanya at MatheMazier.  
If you'd like to sign up to host a future Carnival visit the home site at: http://aperiodical.com/carnival-of-mathematics/

L-l-l-lastly, I'll exploit this opportunity to ask if any math bloggers/writers out there are willing to be interviewed (via email) for my own blog, please let me know: sheckyr[AT]gmail[DOT]com

---------------------------------------------
credits:

(balloon image from AJ at openclipart.org)
(turkey image via Lupin/Wikimedia Commons)
(classroom image via daniel julia lundgren/WikimediaCommons)

Last Call…

This'll be my last call for submissions (deadline is Sun.) to the November "Carnival of Math" being hosted right here at Math-Frolic in a few days. Submission form here:

http://tinyurl.com/krobqjp

(I'd especially encourage newer or lesser-known math blogs to consider contributing as a way of being exposed to a wider audience.)

In other matters, my latest interview is now up at MathTango, with Dr. Noson Yanofsky, author of my new favorite book (literally), "The Outer Limits of Reason":

http://mathtango.blogspot.com/2013/11/noson-yanofsky-limits-of-reason.html

And if you need a little mental workout to start the weekend, Richard Wiseman's Friday puzzle today is a simple Raymond Smullyan-like entry:

http://richardwiseman.wordpress.com/2013/11/08/its-the-fridaypuzzle/

The "Roots of Math Genius"

Physicist and ultimate Platonist Max Tegmark (who argues that the entire Universe is a mathematical structure, composed entirely of mathematics) is one of the promoters of an endeavor called "Project Einstein" which hopes to find the genetic "roots of math genius" by sequencing the genomes of ~400 academic mathematicians/physicists and looking for commonalities. It's a controversial undertaking, you can read more about here:

http://www.nature.com/news/root-of-maths-genius-sought-1.14050

 After noting that one skeptic says, “I thought it was strange that it was called ‘Project Einstein’, which seemed designed to appeal to the participants’ egos,” Peter Woit wryly remarks that "If Project Einstein identifies a common gene among its participants, and uses the knowledge to breed a race of übermenchen, they may find they have selected not for unusual mathematical genius, but for unusual ego." ;-)

Art, Math, and the Realm of Numbers

An interesting (and long) post combining mathematics and art, and "the ability to be deeply intrigued, almost obsessed, by patterns" -- focused in part on "the geometry of irrational numbers, including and especially the geometry of the square root of 2!":

http://www.science4all.org/le-nguyen-hoang/numbers-and-constructibility/

The Centrality of Good Teachers!

As much as I try to stay away (since I'm not an educator myself) from the many controversies and commentary surrounding math education in this country I am constantly lured back to it… and I simply CANNOT EVER NOT read the words of Keith Devlin on the subject since I always find his views so interesting and incisive.
His latest "Devlin's Angle" post is apparently inspired by a piece that educator Liping Ma wrote for the AMS Notices, entitled "A Critique of the Structure of U.S. Elementary School Mathematics," which Keith feels touches on "how to do it right in the curriculum-obsessed, teacher-denigrating US." It also relates to Keith's own recent forays into MOOCs and math video games. Read all Dr. Devlin's thoughts here:

http://devlinsangle.blogspot.com/2013/11/the-educational-power-of-elementary.html

Early on he stresses the importance of excellent, well-trained teachers, over curriculum, with these simple formulas:

BAD CURRICULUM + GOOD OR WELL-TRAINED TEACHERS = GOOD EDUCATION
GOOD CURRICULUM + POOR OR POORLY-TRAINED TEACHERS = POOR EDUCATION

and a couple more brief excerpts:

"Behind Ma’s suggestion, as well as behind my MOOC and my video game (both of which I have invested a lot of effort and resources into) is the simple (but so often overlooked) observation that, at its heart, mathematics is not a body of facts or procedures but a way of thinking. Once a person has learned to think that way, it becomes possible to learn and use pretty well any mathematics you need or want to know about, when you need or want it."

"...it’s very easy to skip over school arithmetic as a low-level skill set to be “covered” as quickly as possible in order to move on to the 'real stuff' of mathematics. But Ma is absolutely right in arguing that this is to overlook the rich potential still offered today by what are arguably (I would so argue) the most important mathematical structures ever developed: the whole and the rational numbers and their associated elementary arithmetics.
"For what is often not realized is that there is absolutely nothing elementary about elementary arithmetic."

I feel like this is must-reading for me… and all the more-so for any educator out there!


...A-a-a-and, as long as I've got your attention, a reminder that the deadline for submissions to the next "Carnival of Math" (hosted right here by Shecky!) is next weekend; so please send those contributions to:

http://tinyurl.com/krobqjp

Improving the 'Dismal Science'

Edward Frenkel's twitter-feed led me to this 19 min. conversation with mathematician Eric Weinstein in which Eric "explores many creative ways that physics and more sophisticated forms of math can be used to rescue economics from itself and restore its now tarnished reputation":

Thursday Potpourri

1) I always forget that Greg Ross's "Futility Closet" site now has an accompanying background blog that runs with it, and which is worth checking out from time to time. When I looked today, it had an interesting little algebraic puzzle, solved by readers:

http://blog.futilitycloset.com/2013/10/07/beef-cuts/#comments

2) The Aperiodical has posted a podcast interview with mathematician/blogger Evelyn Lamb:

http://aperiodical.com/2013/10/all-squared-number-10-maths-journalism/

3) There have been so many wonderful general audience math books out lately I want to again cite four to consider for your math-spending dollars (links given to posts I've mentioned or reviewed them in):

a. "The Simpsons and Their Mathematical Secrets"  -- Simon Singh …been getting lots-and-lots of press attention lately… I haven't read it, but given the subject matter and Singh's writing talents, no doubt it's a good read!

b. "Undiluted Hocus-Pocus" -- Martin Gardner's autobiography …enough said!!

c. "Love and Math" -- Edward Frenkel ...a bit more meat ("Langlands Program") than the above offerings, and not for the math-phobe, but definitely a rich, challenging read for those willing to take a plunge.

d. "The Outer Limits of Reason" -- Noson Yanofsky …I will have much more to say about this book in near future (am reading it now) -- it is quite simply THE BEST math-related book I've ever read, pulling together, as it does, all the sorts of issues I'm most interested in: self-reference, paradox, infinity, logic, uncertainty, epistemology, physics… I hope this volume reaches a much wider audience.
 
And again, a reminder to be sending in those submissions for the November "Carnival of Mathematics" (deadline, Nov. 10):

http://tinyurl.com/krobqjp