Monday, July 28, 2014

Math Education... It's Not Just About Math Content

Jason Rosenhouse posted a great piece last Friday (in his usual succinct style), "Americans and Math," (in response to a NY Times piece that got lots of coverage):

I'm sorry I didn't see more attention given to Jason's post because I think he touches on several points others leave out of the debate over American math education and "innumeracy." He tries to account for why changing math education is so difficult and always has been, including "why it is so difficult to change the habits of math teachers." And he includes some nice examples along the way, as well.

He notes that two of the "big elephants" getting ignored in the debate are the effects of 1) class size and limitations in school resources, and 2) widespread poor respect/support for the teaching profession in general.

In short, the Common Core debate (and Rosenhouse admits he doesn't know a lot of the specifics about Common Core) isn't a simple argument over the best approach to teaching math, but inevitably entails a lot of other variables that ought be part of the discussion.

I encourage all with a stake in this ongoing debate to read Jason's take.

Sunday, July 27, 2014

Sunday Musing

"The fact is, though, that I have come to a point where my wonderment and fascination with the meaning and purpose, if any, of this strange activity we call mathematics is equal to, sometimes even stronger than, my fascination with actually doing mathematics. I find mathematics an infinitely complex and mysterious world; exploring it is an addiction from which I hope never to be cured. In this, I am a mathematician like all others. But in addition, I have developed a second half, an Other, who watches this mathematician with amazement, and is even more fascinated that such a creature and such a strange activity have come into the world, and persisted for thousands of years....

"If you do mathematics every day, it seems the most natural thing in the world. If you stop to think about what you are doing and what it means, it seems one of the most mysterious. How are we able to tell about things no one has ever seen, and understand them better than the solid objects of daily life? Why is Euclidian geometry still correct, while Aristotelian physics is dead long since? What do we know in mathematics and how do we know it?

-- From "The Mathematical Experience" by Philip Davis and Reuben Hersh

Friday, July 25, 2014


From K.C. Cole's "The Universe and the Teacup: the mathematics of truth and beauty":
"There is an old riddle that vividly demonstrates just how noise can interfere with thinking, even when that noise is information. Imagine you are a bus driver. At the first stop of the day, nine passengers get on your bus. At the second stop, two people get off. At the third stop, four people get off, but three new people get on. What color are the bus driver's eyes?" **
And Martin Gardner had his own version of this type riddle (from his volume, "Aha!"):
"You are a taxi driver. Your cab is yellow and black, and has been in use for seven years. One of its windshield wipers is broken, and the carburetor needs adjusting. The tank holds 20 gallons. but at the moment is only three-quarters full. How old is the taxi-driver?" **

**  Since YOU are the bus or taxi driver in these cases, the eye color and age are the same as your own. All the other information is just noise.

(...meanwhile, LOTSA math links compiled over at MathTango this morning for your perusal.)

Wednesday, July 23, 2014

A Few Women of Science...

Sadly, two years ago today, Sally Ride left us at the all-too-young age of 61. I wrote a brief bit about her one week ago.  Her organization for young people, "Sally Ride Science" is here. Lynn Sherr's biography of her came out a little over a month ago. And here's a 1984 PBS Nova clip of Sally as the inspiring, professional, enthusiastic scientist she was:

Meanwhile, this week, Jim Al-Khalili's last seasonal BBC broadcast of "The Life Scientific" (which gets rave reviews) was with distinguished chemist Carol Robinson, about her struggles toward success and recognition as a female in science:

Boys are often encouraged, or even pushed, towards science careers, while girls frequently have similar innate interests discouraged or even blocked.  A couple of viral ads from a month ago addressed that issue, and warrant repeated viewing... though I can only do so with my semi-cynical post from a month ago on the mixture of corporatism and message:

One quick side-note, speaking of daring, independent-minded women, tomorrow marks the birthday anniversary of trailblazing American aviator Amelia Earhart.

What attracts me to these women is not just their abilities, but their courage and determination… especially the courage of their convictions, regardless of societal bounds or expectations, seeking, in a sense, as Thoreau once exhorted, "to live deep and suck out all the marrow of life."

I'll end this by returning full circle to the U.S. Space Program... and John Denver's tribute to the seven astronauts who perished aboard Shuttle Challenger 28 years ago, including the first teacher in space, 37-year-old Christa McAuliffe, a role model for countless young people at the time (an additional female astronaut on the doomed crew was Judy Resnik):

Sally Ride served famously and admirably (alongside Richard Feynman) on the Presidential commission that investigated the Challenger disaster and found decision-making flaws in NASA's organization hierarchy.
I'm not a Ronald Reagan fan, but I'll close out with his famous words to a grieving nation 28½ years ago:
"We will never forget them, nor the last time we saw them, this morning, as they prepared for their journey and waved goodbye, and 'slipped the surly bonds of Earth' to 'touch the face of God.'"

Tuesday, July 22, 2014

2 Requests: Kickstarter + a Survey... and

1)  An emailer asks me to lend support for a kickstarter project (documentary) of some independent NY filmmakers, focused on Erdös-Bacon numbers and networks.

Not sure how much general interest there is for such topics. But Simon Singh did cover them interestingly over a few pages in his popular book on The Simpsons last year. People have had fun with the 'six degrees of Kevin Bacon" as a parlour game for several decades, and of course Erdös is a fascinating character in his own right… so maybe there is a way to draw in a wider audience?
At any rate, see what you think, and pass it along if you know others who might be interested. Whatever brings more math to the public is a positive, and they have less than a month left to make their kickstarter goal.

2)  And in a second emailed request, a survey is being conducted to study the philosophical intuitions of mathematicians "relating to the objects and methodology of mathematics." If you are a mathematician and can take time to participate please go here (they'd like as large a sample as possible, so feel free to pass on to others also): 

Li'l more about it here: 

...Lastly, this might be a nice time to re-link to one of my fave photos on all the Web -- a 10-year-old Terry Tao and Paul Erdös, almost mirroring each other, in contemplation of a math problem: 

...Makes my day!

Monday, July 21, 2014

Li'l Game From Martin Gardner

                   1...  2...  3...  4...  5...  6...  7...  8...  9 

A nice old Martin Gardner puzzle to start the week, which he titled "Fifteen Finesse" in his slim "Aha!" volume (I've adapted it, though still using the same example he used):

There is a new carnival game in town called "Fifteen." It involves a board with the numbers 1 through 9 laid out in order (like above). You play against the carny; doesn't matter who goes first, but you will take turns back-and-forth. You each put coins down one-at-a-time on a single number and then "own" that number ('til someone wins or all 9 numbers are used up). The carny will be putting silver dollars down while you put nickels down. The object is to own any THREE numbers that add up to 15, before your opponent does (for example, 3, 5, and 7). Whoever does this gets all the money played, in cases of draws (no winner) you each take your money back.

So we'll take Gardner's example:
You go first putting a nickel on 7. Carny puts a dollar on 8. You put a nickel on 2. Carny puts a dollar on 6 (and blocks you), realizing that you will win the game if allowed to do so (7+2+6). Now you are forced to put a nickel on 1 to block Carny from winning the game on his next move (8+6+1). Carny puts his next dollar on 4. You again block his chance (6+4+5) at a win by putting a nickel on 5. But next Carny places a dollar on 3 and still wins, since 8 + 4 + 3 is 15. You lose your money. The question is, is there any strategy by which you could be assured a win?

Well, needless to say if Carny is giving you odds of a dollar to a nickel, there's no method for you winning a round… but there is a way for Carny to make sure he never loses a round (wins or draws every game). Can you see it?
What Gardner goes on to explain is that this game is "isomorphic" to classic tic-tac-toe (another game one can always win or draw, and never lose, if played strategically).
There are only eight possible 3-digit triplets that are winners for the Fifteen game:


and these can be arranged in a "magic" square formation:

2 9 4
7 5 3
6 1 8

such that EACH horizontal and vertical line and diagonal is one of the winning triplets.
The carny simply keeps a card illustrating this 'magic square' hidden from (your) sight and makes his number selections referring to it (trying to get 3 in a row, while preventing you from doing so), as if he were playing tic-tac-toe. Then, he can't lose (assuming he doesn't screw up ;-).

Clever Carny, clever Martin.

Sunday, July 20, 2014

Sunday Reverie

Most of you likely know of Steve Strogatz's book, "The Calculus of Friendship," about the touching lifelong relationship between Dr. Strogatz and one of his high school math teachers.
Today, for a Sunday reverie, I'll just link to this 2009 RadioLab episode (15 mins., entitled "Calculove") about the story… worth a 2nd listen if you've heard it before (or already read the book), and definitely worth a first-listen if you aren't familiar with the story:

Thursday, July 17, 2014

Of Course Laplace Predicted Derek Would Do This

The irrepressible Veritasium (Derek Muller) offers a quick lesson on randomness, information, entropy, quantum mechanics, and chaos (in 10 minutes no less!):

One good lay-friendly book that covers/expands on some of this same ground is Charles Seife's "Decoding the Universe."

Wednesday, July 16, 2014

Dick and Sally

Or, maybe I should've entitled this: "Sally Ride... R.I.P."

(Yet another brief diversion today… maybe this will become a series… NOT)

When I posted the Richard Feynman piece last Sunday, an emailer wrote to say (scold?) that I hadn't mentioned a single female in my short list of admired people. So let me rectify that a little bit, with three who come to mind fairly quickly:

First is Jane Goodall, who I admire greatly, but more as a spokesperson and role model than as a scientist (nothing against her, I just don't hold high regard for anthropology!). In the political arena there is Elizabeth Warren who shows tremendous potential, but way too early to tell yet.
The only "scientist" though who really jumps to my mind is Sally Ride (of course there are plenty of female scientists I'm little-familiar with), who died two years ago this month. I largely lost track of her once she left NASA, though I knew she was involved in education (at different levels), and heard rumors of her gender preferences. When she died and the rumors of her personal life were confirmed, I was disappointed. NO, not because of the lesbianism, but because of her failure to publicly acknowledge it during her lifetime. I respect and fully understand that decision, but still her trepidations dropped her down a smidgen for me. It says something VERY sad about our society that someone with the courage to pursue the science/space career she had, nonetheless lacked the courage to simply be herself in public. Pretty clearly, what she feared was the judgment of others; the barbed, shrewish commentary and invasion of privacy that might ensue from culture police, made even worse in the day of the internet.  Sally escaped unscathed, but what anguish did it bring to her life hiding constantly behind such a cloak of privacy...

I can almost imagine Sally and Dick Feynman off in another world now, sitting around a table somewhere, playing poker (...maybe even strip poker at Richard's request!), talking physics and differential equations... and laughing at what a screwed-up world they left behind.

Tuesday, July 15, 2014


Once again, departing from math today (sorry) to help keep this already-viral piece floating along the Web (ICYMI)... Corporate wretch #Comcast doing what comes naturally over the phone (being a corporate ass):

(It's really too bad I don't believe in Hell; I don't get the satisfaction of knowing there's a special place there awaiting some of these folks.)

ADDENDUM: a reader sends me a link to this NSFW vid, so sure, I'll pass it along:

Monday, July 14, 2014

Cycloids (er, Slinkies) Gone Wild

After yesterday's seriousness feel like lightening the mood a bit with this golden oldie I've posted before (there's even some potential math involved, but we'll ignore it):

[BTW, I tried to get back to each emailer earlier today, but some glitchy things were happening with email, so if you emailed me before ~1pm EDT but didn't receive a response yet try again.]

Sunday, July 13, 2014

Richard Feynman... R.I.P.


An entry completely off the beaten track... those who don't care about or like Richard Feynman need not read. This just goes out to his fans at a time he is drawing some heat on the internet:
[...and I'm adding small bits to this post as time goes on, and different issues arise around the Web.]

I've never had many heroes in life; don't even much like the term "heroes," as I tend to see people with all their warts and weaknesses on display. But there are some folks I hold in high esteem. In politics and public life, Ralph Nader, Jerry Brown, Bill Bradley, Morris Dees, and historically, Abraham Lincoln and Clarence Darrow, perhaps Bobby Kennedy if he had lived. I throw the names out, not to name-drop, but just to establish my leanings [I offer a few female alternatives here]. In science, no living heroes. I liked Martin Gardner of course, but probably the closest to a hero, and in a league all his own, was Richard Feynman. I fancied Feynman long before the internet came along (and long before he died), for all the reasons that don't need spelling out to his legion of admirers. As a scientist, a teacher, a thinker, an inspirer, an inquirer, and just a character, he was perhaps unsurpassed in the 20th century.

But, he had his human flaws… he wrote about them; his biographers wrote about them; they were never hidden, the way most of us keep our unsavory skeletons behind closet doors. They were part of who he was, especially as a younger man. And for some reason, now all these years later, they've become grist for some on the internet; pertaining to what we view today as rampant "sexism."
Feynman matured formatively in the 1930s/40s/50s largely in an all-boys network (MIT, Caltech, the Manhattan Project). He was smart, attractive, clever. Did every male in those networks have Feynman's sexist predilections… perhaps not, but I suspect all those with Feynman's charm, intelligence, and hormones did! (admittedly, just my guess)

I'll do a little of what some Webbites patronizingly call "mansplaining":
When new mothers suffer post-partum depression and behave badly around their new-borns, we don't call them lousy, incompetent, dangerous moms (though we could) and recommend they give their babies up for adoption… we try to treat the hormonal/biochemical imbalances they are falling prey to. For that matter, we no longer castigate women suffering major PMS mood disorders and send them off to asylums either, but again try to treat the underlying conditions. Males too, especially young males, and especially young males surrounded by other young males for lengthy periods, are hugely subject to hormonal/biochemical drives in addition to peer pressures. (Of course we are all subject to our biochemistry and neurology throughout our lives.) IF that was the case for Richard Feynman 60-70 years ago, and he lacked certain impulse controls that are expected today, so be it; it does little good to go back and rake over those ashes now. Move on. Nothing more or new to see here. This is all old, old news.

...More old news: My father murdered people 70+ years ago. As a young, wide-eyed, idealistic American he was shipped off to Europe in WWII and ordered to kill Nazis (to 'save the world for democracy') -- don't argue with me about the definition of "murder;" he deliberately killed people, under orders. He came home from that experience a pacifist, having seen atrocities he couldn't fathom… atrocities committed, not just by the Nazis, but repeatedly by American soldiers. I can assure you if your dads fought in WWII or Korea or Viet Nam they saw or participated in atrocities as well, even if they won't speak of it out loud... because, you know, 'War is Hell,' and believe-it-or-not, well, Americans just aren't always choirboys and saints.  But we don't judge our fathers' lives by what they did in such a context, no matter how wretched it might've been. We look over their entire lives, motives, intents, behaviors, contributions, values, and weigh it altogether.

If we were going to judge people of the past by today's standards, then of course all the Founding Fathers fail, just as most of us will fail the ethical standards in place 100 years from now (hell, as far as I'm concerned we are already ethical failures, in addition to being hypocrites and egotists -- a religious person would say we are all sinners, I simply say we are all 'human,' occasionally doing the best we can, usually not). How many of us would wish our entire personal lives opened up to public scrutiny and judgment. The "purists" out there want something that doesn't exist: a scientist with nothing objectionable in his past. I'm quickly judgmental myself of politicians and businessmen (they hold sway over my life and my country, moreso than do scientists), so I understand how easy being judgmental is... but I also know it comes off as sanctimonious and better avoided.

Attacking Feynman to improve the lot for women today, is like attacking George Wallace to improve race relations today; it serves little purpose or impact. Face today's culprits in the here-and-now and deal with them. Or, another analogy, from the corporate world: German transnational company Bayer AG, famous for cropsciences and healthcare, had Nazi associations back in the day (I believe Volkswagen did too???); should we dredge up that history of Bayer and judge them accordingly now? Or is it more appropriate to take companies to task today for what they are actually doing today? (Yo, Comcast...)

Anyway, I'll stack up Richard Feynman's contributions to science, society, education, and humanity, and even stack up his OVERALL values system, against those of his new-born, petulant, self-absorbed internet critics, anytime and have no doubt he'll leave them in his dust. It hurts to see his name dragged through the mud at this late date, especially by people who may never leave any similar lasting mark on the world.

It's a shame that 26 years after his death I feel a need to say Richard Feynman, R.I.P…  Rest... In... Peace.  But we live in a day when it is hubris-driven sport to point fingers at the famous, the vaunted, the iconic, and bring them down to size (and it's especially easy to impugn the deceased). Except Feynman, more than most, already understood his size in the Universe; I hope some other folks eventually understand theirs.

[I apologize to readers for this intrusion into the usual math fare of this blog, but found it necessary. For math fare see my interview with Jordan Ellenberg over at MathTango.]

ADDENDUM: Of course there are really too many to even choose from, but I'm adding one of my favorite clips of Feynman:

[I expect I may be adding more to this post as time goes on. In the meantime I recommend people read the above volume of his letters, edited by his daughter, and reviewed by Freeman Dyson here.]

A few general remarks about some of the stories (which I won't repeat) that have landed Feynman in hot water. First, I'd be cautious about anecdotes from Feynman's own two popular autobiographical volumes ("Surely, You're Joking Mr. Feynman" and "What Do You Care What Other People Think?"). These books were ghostwritten from Feynman narratives, and ghostwriters often embellish, or 'punch up,' stories for marketing purposes. Not all details there are corroborated. Also, Feynman lived through the 'sexual revolution' or 'sexual liberation' of the 60s/70s, a time when it wasn't uncommon for many to broadcast their sexual exploits and freedom… just possibly someone in a field viewed as nerdy as physics might feel compelled to burnish their reputation or prowess a bit with details that today seem appalling.  NO, I'm not making excuses, but am acknowledging that I don't know the precise accuracy of all the tales told around and by Feynman… and, neither do you. For that matter most of us don't even know much about what went on behind the closed doors of our own parents' rooms… that we might find very perturbing in today's 'enlightened' times.

Several of the incidents focused on by critics relate to Feynman's second wife and divorce, a messy affair. The bitterness of the divorce alone mean the exact details as reported must be taken with caution. But moreover, in those days "incompatibility" was not a basis in most states for dissolving a marriage. If one couldn't show "abandonment" (having had no contact with the spouse for 1-2 years) the only alternative to opt for was some form of 'abuse' as grounds for divorce, which again meant that stories were embellished, or even concocted, just to get the legal process moving along.

None of this is to absolve Richard Feynman of his worst behaviors, but the thing is, I don't absolve any of us of our choices throughout the day (in food, clothing, material goods, transportation, entertainment, words, etc.), that send damaging ripples out into the world. All actions have consequences, and we render harm to other people, animals, and the planet, even if unintentionally. All of us (and our parents/grandparents) are vulnerable to character assassination by selective reporting of bits of our lives. I can't stand idly by and watch that happen to an individual who contributed SO much, and who, if alive today, would likely be one of the strongest proponents around for women and minorities in science.

Sunday Reflection...

The domains of mathematics...

"And yet the history of mathematics is a history of aggressive territorial expansion, as mathematical techniques get broader and richer, and mathematicians find ways to address questions previously thought as outside their domain. 'A mathematical theory of probability' sounds unexceptional now, but once it would have seemed a massive overreach; math was about the certain and the true, not the random and the maybe-so! All that changed when Pascal, Bernoulli, and others found mathematical laws that governed the workings of chance. A mathematical theory of infinity? Before the work of Georg Cantor in the nineteenth century, the study of the infinite was as much theology as science; now, we understand Cantor's theory of multiple infinities, each one infinitely larger than the last, well enough to teach it to first-year math majors. (To be fair, it does kind of blow their minds.)….

"Will there be a mathematical theory of consciousness? Of society? Of aesthetics? People are trying, that's for sure, with only limited success so far. You should distrust all such claims on instinct. But you should also keep in mind that they might end up getting some important things right."

-- Jordan Ellenberg in "How Not To Be Wrong"

(My interview with Dr. Ellenberg is now up over at MathTango.)

Friday, July 11, 2014

Laughing It Up With Gauss

Gaussian guffaws....

A few months back Evelyn Lamb introduced me to a humor site I love, called "Gauss Facts" -- a sort of over-the-top jocular fan-site for one of the greatest mathematicians of all time. Worth checking out every few months, or whenever you feel the need to inject a little humor into your day (...and, you're a mathematician):

some example 'facts':

"Gauss can recite all of pi -- backwards."
"Gauss shaves both himself and Bertrand Russell."
"Erdos believed God had a book of all perfect mathematical proofs. God believes Gauss has such a book."

Wednesday, July 9, 2014

Pasch's Theorem/Axiom

Learn something new everyday….

I recently discovered a math misconception I was unaware of… and will pass along for any others not aware of it. "Pasch's axiom" is a little-discussed "axiom," discovered by Moritz Pasch in 1882, that is needed for, but missing from, Euclid's axioms of plane geometry. I'd heard of it previously but now learn that what I'd always considered Pasch's axiom is actually Pasch's "theorem," and that his actual "axiom" is slightly different.

Here is what Davis/Hersh write of Pasch's "axiom" in their classic, "The Mathematical Experience," where I think I may have first learned of it (you can use the line depiction below for visual reference):

"Just what constitutes the 'straightness' of the straight line? There is undoubtedly more in this notion than we know and more than we can state in words or formulas. Here is an instance of this 'more.' Suppose a, b, c, d are four points on a line. Suppose b is between a and c, and c is between b and d. Then what can we conclude about a, b, and d? It will not take you long to conclude that b must lie between a and d.
This fact, surprisingly, cannot be proved from Euclid's axioms: it has to be added as an additional axiom in geometry. This omission of Euclid was first noticed 2000 years after Euclid, by M. Pasch in 1882! Moreover, there are important theorems in Euclid whose complete proof requires Pasch's axiom; without it, the proofs are not valid."
But it turns out that this almost trivial-sounding statement of Pasch's is actually his "theorem," while his "axiom," is stated rather more interestingly and differently:

"In a plane, if a line intersects one side of a triangle internally then it intersects precisely one other side internally and the third side externally, if it does not pass through a vertex of the triangle." (from Wikipedia)

This is interesting in part, because the "theorem" sounds so much simpler or more basic than the "axiom"… reminiscent of Euclid's 5th postulate sounding so much more involved than his other axioms (also interesting that it took until 1882 to discover this missing component of Euclid's axioms). In fact, bopping around the Web, the two Pasch outcomes are often confused, and it is not even clear to me what makes one an "axiom" and the other a "theorem" (and so the confusion is understandable, and not limited to Davis and Hersh).

Tuesday, July 8, 2014

There's Something About Testosterone

Like many, I'm a big fan of NPR's "This American Life," often listening to it's 24-hr./day archival radio station, where I recently heard a replay of their "testosterone" episode from 2002. The whole show is pretty interesting, but I especially had to re-chuckle at a short part I've reported on before. I'll just re-run that post from about a year ago below:


A dash of the offbeat today….

Some weeks back, NPR's "This American Life" ran its 500th show, playing bits from dozens of favorite episodes over the years. Early on, was a 2002 episode piece with an individual who was born female, but always felt more male, and took testosterone shots later in life to finally make the transition in gender. The end of this little segment still rattles around in my mind… After relating a lot of already interesting stories to host Ira Glass about how the change in gender affected him, the interviewee is asked by Ira if there are any other alterations due to testosterone he thinks worth mentioning. The individual responds that after taking testosterone he "became interested in science; I was never interested in science before." To this Ira can't help but chuckle and respond, "NO WAY!" adding that such a response is "setting us back 100 years." The individual goes on to insist that testosterone resulted in "understanding physics in a way I never did before."
You can hear the whole episode here (but the specific exchange occurs around the 22:30 point):

Anyway, what an amazing thought that a few shots of testosterone could, waaah-laaah!, boost science aptitude! I assume math aptitude would be susceptible as well, so I googled "testosterone +math" which led to a 2007 piece [from a blog which no longer exists!] showing some connection.

Toward the end it concludes:

"The important thing to get out of this is that there is an intra-sex correlation between sex hormones and math scores, and this is in the direction one would expect. More prenatal exposure to the male sex hormone testosterone correlates with higher math scores."

before adding this cautionary, anecdotal note:

"However, I also believe that this finding, by itself, does not fully explain the connection between math ability and sex hormones. The  stereotype is that the kids who are good at math are the nerdy non-athletic kids, the opposite of the high-testosterone football playing kids. So although there is a scientifically validated correlation between prenatal testosterone and math, there is an anecdotal opposite correlation for teenage boys. Less testosterone in the teenage years seems to predict higher math performance."

This is just one study of course and there are other studies of the relationship between hormones and science/math ability... and plenty of complicating, intervening variables, as well. Still, it is fascinating that at least one individual, albeit on a pop radio show, feels that the administration of testosterone, like a magic potion, brought on an interest in science that previously didn't exist.


p.s.... for other lovers of "This American Life," if you don't already know this, the 24/7 radio-stream of old episodes can be accessed here: