Thursday, November 29, 2012

"Benny's Rules"

image from Janzak via WikimediaCommons

No need to read this post if you already know all about "Benny's Rules"…. Keith Devlin, fount of wisdom in all things mathematical(!) :-),  recently directed me to this topic, which I was largely unfamiliar with.
Interestingly (to me), even though I was in college in 1973 and quite interested in "learning theory," I don't recall ever encountering the 1973 Stanley Erlwanger paper that introduced the notion of Benny's rules (though Devlin notes it "rapidly became one of the most famous and heavily studied papers in the mathematics education research literature")… perhaps it was more familiar to students specifically in the field of education or even social psychology, but not in the areas of cognitive psychology that I frequented.

It all has to do with the inherent weakness of 'mechanical' means of education, no matter how well-formulated or intentioned (in the instance of Benny, IPI or "individually prescribed instruction" is involved)… i.e. the inability to insure that a student is indeed acquiring a true understanding of the subject matter, even if they are scoring well on automated testing.
The specific focus of the paper is a young student, "Benny," who seems to perform well, and whose teacher believes he is progressing well in his math learning, only to discover upon closer, more personal examination, that he has merely created his own set of rules or patterns that seem to lead to many correct answers, despite a completely false understanding of the actual mathematical process involved. It harkens back to the bottom-line that mathematics is the study of patterns, and the brain naturally searches for patterns… BUT, if not adequately guided or without adequate feedback, a learner may recognize or internalize a very WRONG set of patterns within a given knowledge field.

Read this longish piece first if you're unfamiliar with the Benny case -- especially significant is the final paragraph warning of the potential analogy between Benny's "learning" and that accomplished by a digital consumer of, say, Khan Academy; the author notes "we need to be critical (but not necessarily dismissive) of Khan Academy":

And you can follow that up with another longish post from Keith Devlin from earlier this year:

"...In what rapidly became one of the most famous and heavily studied papers in the mathematics education research literature, Stanley Erlwanger exposed the crippling limitations of what at the time was thought to be a major step forward in mathematics education: Individually Prescribed Instruction (IPI)...

"The subject of Erlwanger’s study was a twelve-year-old boy called Benny, chosen because he was doing particularly well on the program, moving rapidly from level to level, scoring highly at each stage. As Erlwanger states in his paper, Benny’s teacher, who was administering the program for Benny, felt sure that his pupil could not have progressed so far without having a good understanding of previous work.
"Erlwanger’s research methodology was essentially the same as the approach Marilyn Burns used. He interviewed Benny to see what the boy understood. And when he did, a large can of worms spilled out. Though he got high scores on all the question sheets, Benny had almost no understanding of any mathematics, and a totally warped view of what mathematics is, to boot.
"Being bright, Benny had quickly worked out a strategy for tacking the IPI question sheets. His strategy was based in part on pattern recognition, and in part on developing a theory about how the game was constructed – yes, he viewed it as a game! And he did what any smart kid would do, he figured out how to game the game...
"Only when you understand the nature of mathematics does Benny’s strategy seem crazy. Without such understanding, his approach is perfectly sensible. He does not know about math, but he already knows a lot about people and about playing games of different kinds. And when this particular game keeps telling him he is doing well, and making progress, he has no reason to change his basic assumptions."

Tuesday, November 27, 2012

Tuesday Grab Bag...

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

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

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

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

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.)"...

" 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:

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."

Sunday, November 25, 2012

Holiday Shopping... & a question for e-reader fans

Got some mathy folks on your holiday list?… I'll mention 5 math-oriented bookish ideas for your shopping list that I particularly enjoyed from the last year-or-so:

1) The Joy of X by Steven Strogatz
2) The Joy of X by Steven Strogatz
3) The Joy of X by Steven Strogatz
4) The Joy of X by Steven Strogatz
5) The Joy of X by Steven Strogatz

…okay, okay, just joshin' (well, not really ;-))… I'll round out that list with these 5 additional eclectic choices (but with the usual caveat that my tastes/interests won't match everyone's):

2) Measurement by Paul Lockhart
3) One, Two, Three by David Berlinski
4) Proving Darwin: Making Biology Mathematical by Gregory Chaitin
5) The Best Writing In Mathematics 2012 edited by Mircea Pitici
6) The Secrets of Triangles by Alfred Posamentier and Ingmar Lehmann

and (to formulate a list of 10) I'll throw in four other books that I've not read yet, but from reviews and other indications, expect I'd enjoy:

7) The Universe In Zero Words by Dana Mackenzie
8) The Signal and the Noise by Nate Silver
9) The Fractalist by Benoit Mandelbrot
10) Thinking In Numbers by Daniel Tammet

Spend away!

...Now, a question for you Kindle/Nook/iPad/etc. enthusiasts out there: I keep watching the prices drop on these devices and feeling a bit tempted... BUT I ONLY read non-fiction and am not convinced that the reading/learning experience on an e-reader will be that good (especially since most people I converse with buy these gadgets primarily for reading fiction). So am wondering if anyone out there can say what their experience has been (good or bad) reading math/science books on an e-reader?

Saturday, November 24, 2012

Weekend Potpourri

To go along with your leftover turkey sandwiches another potpourri of mathy schtuff(ing) to choose from:

1) First, if you missed it, this nice little demonstration of the Pythagorean Theorem using water as a tool (it's not a technical or real "proof" but a very good 'visual' effect):

2) If you missed NPR's "This American Life" last week (…and you ought NEVER miss TAL), the prologue to the week's stories dealt with Frank Nelson Cole, who back in 1903 proved that a 21-digit number which Marin Mersenne had identified as prime centuries earlier, was in fact a composite number… worth a listen (to at least the opening prologue):

3) Not exactly math, but an interesting British piece on wartime (WWII) cryptography and pigeons flying secret codes; a recently found dead pigeon from the period has a small sheet of code attached to a leg that has stumped modern-day code-breakers:

"...without fresh information this pigeon may well have taken its secret to the grave." The BBC is actively asking for suggestions to decipher the message.

[ADDENDUM: great followup on this story from James Grime here: ]

4) Finally, a good discussion of Bayesian versus frequentist "inference" here, with plenty of comments:

Thursday, November 22, 2012


Skipping the math today….

"If the only prayer you ever say in your whole life is 'thank you,' that would suffice." --Meister Eckhart

Tuesday, November 20, 2012

"The Natural Beauty of Math"

"For mathematicians, however, beauty as an eternal verity has never gone out of fashion. 'Beauty is the first test: there is no permanent place in this world for ugly mathematics,' wrote British number theorist Godfrey Hardy in 1941."
(via Wikipedia)
Read more about Johnson circles and the beauty of math from

And more on Johnson circles here:

Monday, November 19, 2012

Gardner, Gardner, Gardner

But first, here's a nicely-done version of an old puzzle:

'Futility Closet' recently noted that Martin Gardner once did his own variation on this classic puzzle:

And a couple more Gardner pieces recently linked to by Twitterer Colm Mulcahy:

One is the story of Martin's humble work desk:

And a very good, older interview with Martin, by Colm here:

Saturday, November 17, 2012

STEM Yes! Pennies Nyet!

Some pop culture today….

A piece here on Dr. Mayim Bialik of the popular "The Big Bang Theory" TV show wherein she argues for the importance of STEM (including math) education:

It starts off thusly:
"Simply put, The Big Bang Theory actress—and neuroscientist—thinks the United States needs to make science, technology, engineering, and math (STEM) education a priority, and she’s happy to talk to President Barack Obama about it.“'I think when you look at the statistics of which percentage of jobs will require a background in math and science, it’s about 70 percent,' Bialik says in an interview. 'That is speaking to an absolute need for STEM education.'”
...And awhile back I linked to a story about (American) restaurants rounding dining bills up or down to avoid the need for pennies… here's an entertaining (and persuasive) YouTube video that's been around for awhile, arguing for the abolishment of pennies altogether:

Friday, November 16, 2012

Strogatz One More Time

First, if you're sick of hearing about Steven Strogatz here, uhhh, visit someone else's blog today…
Second, just a disclaimer that I have no relation to, nor direct knowledge of Dr. Strogatz (never met the man)… in fact, I have virtually no relatives, friends, or direct acquaintances deeply involved in mathematics at all. ...When I promote folks on this blog it's simply because I'm honestly impressed with their math communication efforts.

With all that said... again today... a link to another recent piece centered on Dr. Strogatz, wherein he offers "7 creativity tips" for mathematicians and others, including the value of "loose and amorphous" (versus "rigorous") thinking, especially since "intuition is real." The tips are wonderful enough, but the post then ends with a fabulous (8-year-old) TEDTalk from Steven that I was unaware of on another of his favorite topics, synchrony (it's 21 mins. long, so try to make time for it):

And here's a little improved example from YouTube of the metronome synchronization demonstration that Dr. Strogatz uses in his talk:

Thursday, November 15, 2012

Facebook Math

I just stumbled across this Facebook page with some fun mathematics:

I actually don't spend much time with Facebook, but if anyone has other favorite entertaining or interesting math-related pages there please let us know in the comments.

Tuesday, November 13, 2012

Doing The Thing You Love...

Just a couple of videos today pertaining to the love of math....

A few days back I linked to a brief audio of Steven Strogatz speaking at the American Academy of Arts and Sciences, and now a YouTube video of the ceremony has been uploaded, so once again enjoy Steven spending a few minutes talking about doing what he loves doing (...and how quantum mechanics and an emancipating mother guided him there):

Not sure exactly how old this 2nd video is (or if I may have linked to it in the past), but another wonderful older offering from the BBC Horizon series, entitled "A Mathematical Mystery Tour"(covers a lot of ground in 50 mins.):

Monday, November 12, 2012

Ignore Bayes At Your Peril

...are you listening, Karl Rove?

Thanks to the success of Nate Silver in projecting the U.S. Presidential election results, Bayesian statistics have been getting a lot of internet play of late. Bayes' Theorem "bridges the gap between probability and logic" by looking at events not solely in terms of randomness, but in terms of updated known information that can be factored in (probability becomes "conditional" instead of simply random). The idea is simple though the actual application can get rather technical. Here's a basic introductory post to the subject from Grey Matters blog (which includes several additional links and videos):

Plenty more on Bayes' Theorem from a Google search:

...and on YouTube as well:

...And just today 'Mind Your Decisions' blog posted about a simple medical diagnosis problem, which hints at the potential pitfalls of a purely "frequentist" (non-Bayesian) approach to statistics: 

Sunday, November 11, 2012

An Interview with TERENCE TAO...

(via WikimediaCommons)
...well, okay, not by me :-( (maybe some day), but wonderful nonetheless. I stumbled across this 2009 interview with Dr. Tao from an Australian mathematics gazette, and (though I often avoid linking to pdfs) it's tooooo good not to pass along; many great responses from Terence (and no math required!):

Here's one interesting little Q & A exchange:
"Gazette: What was the best career advice you have ever received?

Tao:  Mostly people have led by example other than explicitly giving advice. I do remember one thing my advisor told me once, which was very useful. I was writing my first paper, and I put a little joke in it. I thought I was being smart. He took a look at me and said: ‘When you write a paper, this is something that will stay in the record for ever. Thirty years from now people will still read it. What you think is funny now, may not be funny thirty years from now’. He told me not to put jokes in my papers. Looking back, that was actually pretty good advice: don’t be a smart alec when you write. And it wasn’t a very good joke anyway."
And here's a second (non-pdf) interview with Dr. Tao from the Web (not as good as the above, but still good):

If perchance you're not familiar with him, Dr. Tao (Wikipedia page HERE) is one of the world's foremost mathematicians (a former Australian child prodigy of Chinese ancestry), a Fields Medalist, and currently professor at UCLA. A Smithsonian Magazine article on him here:

And his own mathematics blog here (which on rare occasions includes posts even I can understand! ;-):

Finally, another fascinating (and longish) piece below on Terence, his two gifted brothers, and the parents that raised them. It delves into both gifted and autistic children and is, I think, a super-read:

An excerpt:
"Billy Tao [the father] recalls that early on he and his wife were helped by the Gifted and Talented Children’s Association of South Australia, but they became uncomfortable with the fixation some parents in the association had with their children’s achievements and IQ ratings. The Taos had already seen the pitfalls of this approach during a trip to the US when they met Jay Luo, a prodigy who had earned a university science degree at age 12, but later dropped out during his PhD studies.
"'Many parents of gifted children tend to overestimate their children’s ability, they want to maximise speed,' says Billy. 'One thing I disagree about with the gifted-children movement is the emphasis on acceleration. Many gifted-education people, particularly teachers who have diplomas in gifted education, are all brainwashed with this idea of acceleration, acceleration, acceleration. What about lateral thinking? What about creativity?'
"In contrast to the effusive praise other parents heaped on their little Mozarts, the Taos avoided excessive flattery and downplayed the importance of winning."

[p.s... sorry to sound like a broken record, but if anyone (math blogger) is willing to be interviewed here for my series pleeease don't hesitate to let me know that -- it's far more efficient for me to contact people who have already consented to the process than to blindly send out inquiries and not know whether they'll be returned or are even received. I really like throwing a little extra attention to folks who are out there actively promoting math to a wider audience.]

Thursday, November 8, 2012

Greg Ross of Futility Closet

Math-frolic Interview #8

"An idler's miscellany of compendious amusements." --Futility Closet

Another major treat for me today! If the name "Greg Ross" doesn't immediately ring a bell, you might still know his blog "Futility Closet," one of the most consistently entertaining blogs on the Web for geeky types! Despite it's relatively simple, plain presentation, it's become a daily stop for 1000's of readers.  Futility Closet is not a math blog per se, but does have its share of math content (and puzzles and chess problems also), and it was great fun for me to learn more about its proprietor (...oh, and learn where the name "Futility Closet" came from also)! Enjoy....


1) Most folks I interview here are math bloggers, but your blog "Futility Closet" is different and hard-to-define. I know you work professionally as a scientific editor, but what is your academic past… do you come out of a journalism or writing background, or more of a science-trained education (or something else)?
My training is in magazine journalism, but I've spent most of my career working with engineers and scientists, first at IEEE and now at American Scientist. And I've spent some time in education as well, at the National Educational Service and then at, an online university.  I'm happiest when I'm learning, but there's no one discipline that captivates me. I suppose that's reflected in the blog.
2) Can you say how math fits into your life and work… is it just one among several side interests, or anything more than that?
It's just one, really. I love the beauty of math and the ingenuity of mathematical arguments, but I'm equally drawn to history, philosophy, and language. I guess I'm attracted to interesting ideas generally, and math is certainly a rich source. The blog has given me an excuse to explore math a bit more deeply than I otherwise would have, but I'm definitely an amateur.
3) Your blog is one of the quirkiest ones out there… and yet it becomes a favorite for everyone I know who finds it. How did the idea for the blog first come to you, and how confident were you that there would be an audience for it? Also, where does the title "Futility Closet" originate from?
When I started at American Scientist I wanted to create an informal site where I could experiment with web development. The content was really an afterthought -- I'd experimented with blogging in the past, but writing about my own opinions seemed too self-involved for me. So I just tried to envision a site that I myself would want to read, following O. Henry's dictum "Write what you like, there is no other rule."  For me that meant a collection of concise, self-contained posts on curiosities in history, literature, art, science, philosophy, and math. O. Henry knew what he was talking about -- I've never promoted the site, but it's getting a million pageviews a month now.
The name Futility Closet was sort of an accident. My wife went to American University in Washington D.C., and one day while visiting there I saw a door marked UTILITY CLOSET where someone had scratched an F into the wood next to the placard. I'm sure there's a great story behind that, but I don't know what it is. Years later, when I wanted to buy a domain for the site, that was one of the few names I could think of that wasn't already taken. I suppose it's memorable, at least.
4) Do any particular math posts that you've done come to mind as personal favorites? And how about puzzle posts?
I think the most popular math post of all time was the one on spirolaterals, from April 2010:
That involves a bit of a mystery. I first came across spirolaterals in The Penguin Dictionary of Curious and Interesting Geometry, by David Wells, who gives the swastika example but not its source. Wells credits Frank Olds for inventing spirolaterals and Martin Gardner for discussing them in his November 1973 column in Scientific American, but none of these three explains who discovered the swastika example, so I don't know whom to credit for it. It's certainly striking.
This summer Gary Antonick at the New York Times asked for my favorite puzzles -- I sent him these:
I told him that I seem to be drawn to puzzles that have a mathematical character but that require little math to solve; that's what these seem to have in common. He was asking because Mark Frauenfelder had run the "catbird seat" puzzle on Boing Boing, and Gary picked that up in September:
5) Approximately how much time per week do you spend working on your blog? And is it principally "a labor of love" or something more than that for you?  Might there ever be a "Futility Closet" book in the future?
I guess it's a couple of hours a day now on weekdays, and then flat-out on weekends.  My day job is demanding, and I have freelance work to do as well, so the blog takes up what used to be my free time -- it involves a lot of trips to university libraries. I'm sort of dog-paddling toward developing the site further, first by redesigning it and then, hopefully, starting a series of books. There seems to be a lot of potential, but it's a real struggle to find time to work on it.
6) You deal with a lot of real oddities on your blog… you're probably a pretty skilled fact-checker(!), but have you ever published a post only to find out later its content was inaccurate, fake, questionable, or for some other reason ought be retracted as published?
Yes. I spend about half my research time in fact-checking, but even then errors get by. Happily the readers are pretty erudite, and they tend to put me right when this happens. The most recent case occurred last month with a post on the paradox of the second ace ( I had published that originally in February 2009, but a reader showed me that I'd flubbed the setup by letting the player volunteer the contents of her hand, rather than answering questions put to her by an observer. So I took it down and it sat in my notes for a year until I found the time to work on it again. I think it's okay now. :)
7) Do you from time-to-time repeat posts you've done before, and if so, do you have a rule-of-thumb for how much time must pass before you'll re-run post material used previously?
No, I try not to do this -- it would mean deleting the older post, which would break any links that readers had made to it. Plus it would be boring for long-time readers, who have already seen the material. I'll make an exception if an old post can be developed in some way or combined with new material to make something that I think warrants renewed attention -- the paradox of the second ace is a good example. But generally I try for fresh material with each post.
8) You don't have "comments" on your blog-posts… is that just to avoid the headache that they can be or for some other reason? And do you get a lot of email feedback from readers?
I correspond with readers all day, but I think general experience has shown that a good Internet community requires pretty close moderation, and at the moment it's everything I can do just to write the content.  But that's one of the developments I'm considering -- I know from my email correspondence that we have the makings of a good community, if that's what readers want, either post-specific comments or a general forum.  We're redesigning the site now, and the new version will include an author blog where we can discuss what to do (if anything).
9) To round yourself out a bit, when you're not researching/writing geeky things, what are some of your main interests/hobbies/activities?
Oh, man, nothing. I run, and I used to play guitar and piano, and chess, but right now I'm just working flat-out all the time. What a terrible answer!
10) Any parting words, not covered above, you'd care to pass along to a math-oriented audience?
No, except to thank them for their interest and their patience with me. I'm always looking for material, so if anyone has a prospective submission, they're welcome to send it to me via the site. Thanks for reading!

...If you're not familiar with Futility Closet definitely check out some of the links Greg has passed along here (you'll get hooked)... oh, and if you are familiar with Futility Closet RE-check out, and RE-enjoy, some of the above classics that Greg has provided (and you can always search his archives via several categories as well).
Thanks Greg for finding the time to take part here; great fun for me. Looking forward to the books!

Wednesday, November 7, 2012

Mother, Country, and Apple Pie, er, uhh, Mathematics

(via his Cornell page)
What is there not to love about Steven Strogatz… in the popular math arena, he is becoming, like Martin Gardner and Raymond Smullyan, another American gem. In this short (7 min.) speech before the American Academy of Arts and Sciences he wears not only his love of math on his sleeve… but his love of mother and country as well:
(it's the 4th audio segment down on the page)

[p.s...A quick note of thanks to those who've sent in suggestions for interviews here, and tomorrow morning I'll have another from my original list of folks that I really think readers will enjoy, and yet don't imagine you'd guess ahead of time who it is (no, it's not Steven).]

A Bit of Serendipity

Since the prior post was heavy with some Tim Gowers material I can't help but point out another cosmic synchrony ;-) that materialized on the Web… A short while back Patrick Honner (who I just interviewed recently) posted this note on his blog:

...basically, he notes that a purported sine wave curve on a NY State math exam simply didn't look right to him at a glance ("too rounded"), and then goes on to show that his visual hunch was indeed correct upon closer inspection. The ensuing comments to his brief post are fascinating as well, but the capper is that about 10 days later, in a bit of serendipity, Tim Gowers writing on Google+ noted his immediate sense that a British air traffic control tower he'd passed by recently did not conform to a "mathematically natural shape," and wondered aloud "how acute this sense that mathematicians have is," at which point he stumbled upon Honner's post:

…it reminds me a wee bit of the old retort (applied when you succeed at something, and someone else says it was 'just luck'), that it seems 'the more I practice the luckier I get!' ...or, in-other-words, it's a skill developed from experience.
Having said that though I s'pose it's a bit difficult to know which is the cart and which is the horse here: i.e., do mathematicians have a special knack for recognizing certain patterns and forms because it's something their minds have experienced repeatedly (practiced)… or, is it that people who have a special or inborn aptitude for form and pattern recognition are more likely to end up in mathematics???

[p.s... if anyone hears the result of Tim Gowers' recent heart surgery and how he is doing, please report to us in the comments; I'm sure many would like to hear that all went well.]

ADDENDUM 11/10/12: Tim now has a detailed post up that he is back home recovering (with some discomfort) from the hopefully successful surgery:

Tuesday, November 6, 2012

Mochizuki, Gowers, Alan November

First, a wonderful article from The Boston Globe emphasizing how difficult it will be to check Shinichi Mochizuki's lengthy claimed proof of the ABC conjecture, because of the complexity and newness of the math involved:

The piece ends thusly:
"And this, [Minhyong] Kim thinks, might pose the greatest challenge of all. 'When you’ve been wrapped up in your own research program for a long time sometimes you lose a sense of what it is that other people don’t understand,' he says. 'Other people feel quite mystified as to what he’s [Mochizuki] doing and part of him, I suspect, doesn’t quite understand why.'"

On a simpler note, do you wish to engage your own math students… well, so does Tim Gowers...
British Field Medalist Gowers (who deserves to be read/discussed whenever possible) has a piece in The Spectator regarding math education:

from it: "...rather than explaining mathematical ideas (about statistics, say) and then discussing how they can be applied to the real world, a teacher should instead start with a question that is interesting for non-mathematical reasons and keep a completely open mind about what mathematics has to contribute to the discussion."

Gowers' take is that teachers need to be utilizing real 'real-life' examples in the classroom, not the 'if 2 painters can paint 2 houses in 5 days how many houses can 4 painters paint in 15 days' sort of story problem that often gets passed off as an application.

This is more-or-less a followup to a fantastic earlier piece he did on same subject:

[On a side note, just yesterday, Gowers put up another interesting post about probabilities and surgery he is undertaking for an atrial fibrillation condition… hopefully, we'll soon hear that all went well!]:

Finally, hat tip to John Golden for leading me to this TED video (~16 mins.) by Alan November that piggy-backs nicely onto Tim Gowers' views, in describing student engagement in learning:

Sunday, November 4, 2012

Sol Lederman is Wild About Math

 Math-frolic Interview #7

"Making math fun and accessible" -- Sol Lederman (from his blog)

Another delight for me today... I suspect anyone reading my blog is well-familiar with Sol Lederman's "WildAboutMath" blog -- it's been around a long time, and makes virtually every list of 'best math blogs on the Web' that ever comes out! Sol was actually the first person I asked for an interview, but being a busy guy, later requests to others actually got back to my email box faster. But better late than never!

Sol loves talking with people who are "inspired by math." I'm not sure though that he realizes just how inspiring HE has been to many of us in the blogosphere… I would never have attempted a simple and sometimes light-hearted math blog aimed at a general readership, had Sol's "Wild About Math" blog not blazed a path showing there was an audience for a not-too-technical approach. Moreover, his recent podcast series was the inspiration for the interview series here.
So without further delay ....he-e-e-e-e-ere's Sol:


1) To start, could you tell readers a little about your background or anything else pertinent to your becoming a math blogger…

I've loved math forever. And, I enjoy writing. Some years ago I became burned out on my techie work and wanted to make a change. So, I jumped ship and started blogging. That was five years ago. I didn't stay away from tech for very long, though. I'm back doing programming, tech support, and other related things. But, I still blog occasionally although far far less than I once did.

2) Beyond writing a blog, where does math fit into the rest of your life or work? (if not already covered above)

Math for me is about logical thinking AND about creative expression. Depending on the task, I sometimes get to do creative thinking at work. Certainly the mathematical qualities of problem solving, pattern matching, associating one idea with another, and attention to detail carry over from my math to my work.

3) What are your favorite aspects of mathematics (that you yourself most enjoy studying)?

I'm very intrigued with the question of how to make math more understandable to the masses. Just today I was having a conversation with my boss about how fractions work and why the techniques we have for manipulating fractions actually work. Why is it that we can multiply fractions? What does it mean to divide a fraction by another fraction? How much of this is pure abstraction vs. grounded in something that students can understand and touch? Anything that demystifies math is exciting to me.

4) "Wild About Math" has been one of the most popular and consistently-highly-rated math blogs on the Web (it recently had it's 5-year anniversary!). Can you tell us how it first came about in your mind, and how confident were you that there would be an audience for it?

I once thought that I'd like to lead workshops with kids where we would explore the "wilds of math." I did do math circles in Santa Fe for nearly a year but I never did create the workshop for kids. But, the idea for a name stuck, and became Wild About Math!

5) Are there any funny or entertaining behind-the-scenes stories you can tell from having run a blog for 5 years?

Early in the blog's history I wrote a post, Impress your friends with mental math tricks. I wrote that post quickly and without a huge amount of thought or editing. That post got 45,000 views one day, mostly from folks at StumbleUpon. It made the front page of Digg and put the blog on the map. And, that humble post continues to bring traffic five years later. No other post has ever come close to achieving that status!

Me: ...WOW!, 45,000 that's amazing… I was thrilled in the very early days of my blog when YOU once linked to a puzzle I presented and I immediately got several hundred additional visitors, and as you indicate, I still today get visitors to that post every single week from your original link. So I hear what you're saying even though I've never experienced it on that scale.

 6) Are there certain blogposts you've done that stand out for you as personal favorites or ones that were the most fun to work on? And from the other side, which posts seem to have been most popular or attention-getting with your readers?

This fun post is the second most popular article from the blog. It gives 8 simple math-based games that kids can play with paper and pencil. My favorite posts are the recent podcast interviews I've done with people making a difference in math education.

7) Name some of your favorite math books that you like reading for enjoyment, and/or books that you recommend to people who are interested in math but may lack a strong academic background in it.

I recommend any of James Tanton's books, books by Theoni Pappas, Keith Devlin, and the late Martin Gardner, of course.

8) You're very interested in math learning and education. What's your view of Khan Academy and the ongoing controversy that seems to surround it in some quarters? Are there online instructional math sites that are your favorites? And have you taken any online courses yourself (perhaps Keith Devlin's latest offering)?

I've not taken online math courses and I'm not impressed with most instructional math sites. Regarding Sal Khan's work, I feel he's doing the same old paradigm, lecturing to students, but on video instead of in a classroom. I am VERY excited, though, about the work of Scott and Jen. They are doing phenomenal stuff with online game-based education. They are the only online site I'm excited about.

Me: Interesting, hope readers check it out...

9) For awhile now your blog has focused on podcasts with individuals who, as you say, 'are inspired by math, and inspire others'. Is that the format (podcasts) you see the blog taking for the foreseeable future, or any plans to return to more written posts at some point, or do you have any other vision for the future of your blog? And can you say anything about which 'math inspirers' we may look forward to in future podcasts?

For the foreseeable future, yeah, I plan to keep doing podcasts. I like doing them much more than writing book reviews. Princeton University Press and the MAA send me a fair number of books that I really like so you can expect to see me interviewing their authors. But, I'm open to interviewing a wide variety of authors. I feel like my momentum for math blogging has been slowing down. I've got other interests I'm pursuing so I don't expect to do much more than the monthly or so interview that I'm doing these days.

10) Any other parting words, not covered above, you'd care to pass along to a math-oriented audience?

I sincerely appreciate your interest in interviewing me and I've said more than once that I really enjoy your blog. I don't read math blogs very often these days but when I'm in the mood to scan a few I always make it a point to check out yours.

Me: Thanks for the kind words; much appreciated, and great to know you're a fan! 


Sol further asked about my availability for a podcast on his blog, but for now, I think he has too many really interesting, knowledgeable people to choose from before I take up his time! Maybe one day though....

Thanks for participating here Sol, and congratulations on 5 great years of successfully making math fun and accessible!! (And I've added a permanent link to WildAboutMath podcasts in my right-hand column "podcast" listings.)

Saturday, November 3, 2012

Math and Writers...

An interesting (if a bit esoteric) essay in the current New Yorker on "Why Writers Should Learn Math":

some quotes from the piece:
"What ballet is to football players, mathematics is to writers, a discipline so beguiling and foreign, so close to a taboo, that it actually attracts a few intrepid souls by virtue of its impregnability. The few writers who have ventured headlong into high-level mathematics—Lewis Carroll, Thomas Pynchon, David Foster Wallace—have been among our most inventive in both the sentences they construct and the stories they create."

"As the mathematician Terence Tao has written, math study has three stages: the 'pre-rigorous,' in which basic rules are learned, the theoretical 'rigorous' stage, and, last and most intriguing, 'the post-rigorous,' in which intuition suddenly starts to play a part."

"If Hemingway’s writing is algebraic in its precision, then [David Foster] Wallace’s is quantum calculus..."
and a couple of other quotes included in the article:
 “A mathematician, like a painter or a poet, is a maker of patterns... The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test.” --G.H. Hardy

"Those who’ve been privileged (or forced) to study it understand that the practice of higher mathematics is, in fact, ‘an art’ and that it depends no less than other arts on inspiration, courage, toil, etc.” --David Foster Wallace

Friday, November 2, 2012

As Conan O'Brien Might Say…

…'Where are the la-a-a-adies???'

I'll be posting Math-frolic Interview #7 on Sunday, and while I'm thrilled with the guest, it is the 6th time out of 7 with a male interviewee (and the majority of inquiries that I'm waiting to be returned are also guys)… soooo, a little more representation of the Emmy Noether gender would be nice. 

One problem is simply that most math bloggers are likely male, and secondly I've discovered that many female bloggers don't as readily post their email addresses, making initial contact difficult.  There are likely a number of female math bloggers/teachers though that I'm just not very familiar with; so again, please feel free to make recommendations if you think someone would make a good interview.
…Oh, and don't everyone mention Vi Hart at once -- of course I'd love to interview Vi but suspect she may be toooo busy at the moment to squeeze it in (though I did attempt to send her an email, which she may or may not have received?). But after Vi, who else?

Meanwhile, for some reading entertainment (especially if you're going out to eat this weekend), an interesting little story that made waves a couple months back, about the 'rounding' of restaurant bills up or down to do away with the need for pennies (the story focused on Chipotles, but it happens elsewhere, and there can be pros and cons):

Thursday, November 1, 2012

'Best Of' Math 2012...

A quick blurb about "The Best Writing On Mathematics 2012" which I received a review copy of. It is once again edited by Mircea Pitici, and I believe this is the best edition thus far. Upon reading the first edition in 2010, it seemed a wonderful idea, the execution of which would improve in coming years. The 2011 volume was better than 2010, and this year's version has hit its stride even further, especially in containing more pieces that should appeal to a wider audience.
There are several well-known names (Terry Tao, Tim Gowers, Mario Livio, Brian Hayes, John Baez…) in this edition. The Foreword by David Mumford is a wonderful delineation of pure versus applied mathematics (also throwing physics into the mix). And I enjoyed most all the early entries in the volume. It slogged along a bit more in the middle and toward the end (for me), and, like most anthologies, overall, contains a mix for many different tastes. I doubt that anyone will enjoy every single entry, but that is in the nature of such (somewhat disjointed) compilations. At the end of the text, editor Pitici gives a list of other articles considered for inclusion that didn't make the cut. Of course several of them look quite interesting as well, although I don't know how readily available they are for lay readers.

There may be fewer technical pieces in this edition than previous ones, and I'll warn readers there is a lot of material on philosophical and historical underpinnings of mathematics -- topic areas I find rich and enjoyable, but some do not, and will be less enamored than I of certain portions of the book. There is still some fairly technical reading involved as well, so I'll also warn that this remains largely a volume for those already inclined toward mathematics and having some preparation; it is not for the general reader hoping to be newly engaged by the "best writing" to be found on mathematics. Unlike, say the 'best science writing in 2012' which might still appeal to a non-scientist, large chunks of 'the best writing in mathematics' will still only be accessible to math buffs.

The two oddest or quirkiest offerings (to me), came near the end: an essay by Fernando Gouvea on whether or not Cantor was truly surprised (and over what) when he famously wrote (in translation), "I see it, but I don't believe it" (essentially upon proving that a line segment contained as many points as a multi-dimensional geometric form), and the very last piece by Mark Colyvan on mathematics and dating/mating algorithms.

My favorite piece also was toward the end... I only discovered Brit Richard Elwes a couple years ago and he has quickly vaulted forth as one of my favorite math explicators. His essay bringing the subjects of Cantor set theory, infinity, and the Continuum, up-to-date with current ideas of an "ultimate L" logic-world promoted by Hugh Woodin is excellent and mind-bending. I'll be going back to re-read his and several other of the entries more slowly/carefully when time permits, and looking forward to next year's edition as well! For now, this series seems to be Pitici's 'baby,' and I don't wish to take anything away from his efforts, but it would be interesting to see what different editors each year would come up with (as is the practice with the more famous annual "best American writing" series that's been around for awhile).
Despite what the public might think, math is an incredibly diverse field of study, and therefore difficult to anthologize. If you're not already acquainted with Pitici's series I recommend this 2012 volume as a place to start.