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Sunday, February 28, 2016

Sunday Reflection, er uhh, Chuckle


I was reading some old John Allen Paulos this week, and these two gems popped out at me... not mathy, but too good not to pass along:
"...two clergymen discussing the sad state of sexual morality.
"'I didn't sleep with my wife before we were married,'  one of them declared self-righteously. 'Did you?'
"'I'm not sure,' said the other. 'What was her maiden name?'"
...and, quite different, (but on the very same page) this:
"What is a question that contains the word cantaloupe for no apparent reason?"
;-)
-- all from John's book "Beyond Numeracy"


[p.s. -- please visit MathTango today for a new Math-Frolic Interview!]


Thursday, February 25, 2016

Blast From the Past... the Quincunx

via Svjo/WikimediaCommons

Almost six years ago I posted about one of my earliest memories of something mathy that mesmerized me. I was interested in math and numbers at a young age, but my first view of a "quincunx" or "Galton board (or box)" was something that touched me at a deeper level.

So just a little nostalgia today as I re-run most of that earlier post below:

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Harking back to my childhood today....

When I was a youngster visiting a science museum in my home state what most fascinated me was not the dinosaur displays, or fossils, or insect collections, nor more whizbang exhibits, but a simple large display known as a "Galton Box" (after the 1889 inventor of the first one), or known by some as a "quincunx," (...okay, so I was an odd kid).

Most of you are likely familiar with these enclosed contraptions in which balls drop from a central point at the top onto a symmetrical pegboard where they bounce around until finally falling into columns at the bottom... the majority of balls dropping, by sheer chance, somewhere in the middle columns, and fewer balls bouncing around in a manner depositing them to the outer end columns.

On the glass pane enclosing the balls and peg-grid would be drawn the 'normal' or 'Gaussian' distribution (or 'Bell curve') so central to mathematics/probability, and lo-and-behold, once all the hundreds of balls had been released they would, in the columns below, take on the shape of that normal curve, via of course the 'laws' of sheer chance, not due to any mechanical manipulation. Even as a child, not really understanding much about normal distributions, nor math/probability more generally, somehow that demonstration was very powerful to me; like a magic, unseen hand guiding the fate of those individual spheres ---  each one taking a rather random, unpredictable journey, yet the end result being highly predictable and little-changing. Even as a youngster I sensed there was something profound in that. Some kids today construct Galton Boxes or quincunxes for science fair projects. Hooray!, for still to this day I love these apparatuses and their magical outcomes (...the rest of you can go gawk at dinosaur models).

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Here's an animated version of a Galton Board from the Web:

http://www.mathsisfun.com/data/quincunx.html

And here's some chap you may recognize ;-) demonstrating a real-life miniature one:



...and finally this one, made from Legos, may give you some ideas if you wish to construct your own:
https://www.youtube.com/watch?v=aGiKdJ2npc4


Tuesday, February 23, 2016

We All Love Lists, Right...


Not much math here, but I'm intrigued a bit by philosophy of science (as an outside observer), so perhaps some of you are as well. A h/t to physicist Sean Carroll for recently pointing out this compiled list of "best Anglophone philosophers of science since 1945":

http://civs.cs.cornell.edu/cgi-bin/results.pl?id=E_81c68ea1c38fe926

Admittedly, I'm not even familiar with 2/3 of the 100+ names on this list (including "Bas van Fraassen," #5 in the rankings), but was still surprised by a few things about it:

1)  I enjoy reading Deborah Mayo's "Error Statistics Philosophy" blog, and so was unexpectedly pleased to find her ranked 59th out of 104 on the list.

2)  Despite the "popular" appeal of Thomas Kuhn's writings, surprised to see him at #2 in this list (thought his aura had faded somewhat, but apparently not).

3)  Also, didn't realize that Carl Hempel was viewed this highly -- could have anticipated a top 10 showing, but #3! and a notch ahead of Karl Popper at #4 would not have guessed. Feyerabend also higher than I would've expected (at #9).

4)  Surprised too that David Bohm even appears on this list (at #53), and the ubiquitous Wittgenstein does not appear at all (I assume his death in 1951 precluded inclusion here?).
But biggest surprise of all was Rudolf Carnap winning the poll fairly handily. He too (and "logical positivism," more generally) I thought had well-declined.  Apparently Martin Gardner was correct in predicting, years ago, a resurgence for Carnap (one of his professors), given some time! (a short piece by Gardner on Carnap is Chapter 3 in his great volume, "Are Universes Thicker Than Blackberries" HERE.)

Anyway, interesting compendium... any surprises for other readers?

Monday, February 22, 2016

Circular Thinking ;-)



I've read about "math circles" in the past but only recently observed them in my local community.  Young people (K-12) with a penchant for math attend math circles (usually once-per-week) to be challenged by more enriched experiences/problems (not tutored) than they likely get from their classroom experience. 
Math circles have been around for decades in the U.S. (having started overseas) and may take on slightly different styles or formats. I feel safe in saying the overall goal is to give young people greater practice in mathematical thinking, although competitions and computational techniques can sometimes be a part of the process. And many now have long waiting lists to attend, such is their popularity.

I've been impressed watching the joy, enthusiasm, and thought (from both students AND teachers) of math-circle participants in my local area.
Read more about them here from Wikipedia:

https://en.wikipedia.org/wiki/Math_circle

...and here from an official website:
https://www.mathcircles.org/


(there's also a Facebook page for math circles:  https://www.facebook.com/MathCircles/info/ )

Check below to see if there are any established math circles in your area:
https://www.mathcircles.org/Wiki_ExistingMathCirclePrograms_view

Lastly, by coincidence, Cathy O'Neil recently ran a guest post on the Chicago math circles at her Mathbabe blog:
http://mathbabe.org/2016/02/12/guest-post-a-math-circle-thats-breaking-the-mold/



Sunday, February 21, 2016

Rudimentary vs. Complex


Not particularly mathy, but a bit of Alan Watts' mysticism this Sunday:
"All I'm saying is that minerals are just a rudimentary form of consciousness, whereas the other people are saying that consciousness is a complicated form of minerals."

Thursday, February 18, 2016

Flipping


Contemplating 'flipping' your classroom?... many are, or already have...some recent links:

Stephen Cavadino's positive experience implementing the "flipped classroom":
https://cavmaths.wordpress.com/2016/02/13/flipping-the-classroom/

and Dan Rodriguez-Clark here:
http://www.interactive-maths.com/blog/third-year-of-flipping

Meanwhile, Robert Talbert, is writing a book on the flipped classroom approach, and in this post delineates what he views as the 3 main reasons for choosing a flipped mode:
http://rtalbert.org/blog/2016/three-reasons-for-flipped-learning

A couple more (of MANY) sites/blogs with videos/resources for flipped classrooms:

http://www.edutopia.org/blogs/tag/flipped-classroom
http://jonbergmann.com/blog/
...and from YouTube:
https://www.youtube.com/results?search_query=%22flipped+classroom%22

(am just passing these along, but haven't perused them enough to judge their quality)


Wednesday, February 17, 2016

Of Averages, Standard Deviations, Confidence Intervals, Probability


Worth noting perhaps that "Statistician" is one of the fastest-growing occupations in the American workplace:
http://thisisstatistics.org/statistician-projected-as-top-10-fastest-growing-job/

Some stat-related pieces catching my attention recently...

1)  Starting with the simplest, Todd Rose looks at learning and "averages" in this interview from NPR (...none of us are "average"):
http://tinyurl.com/jeryann

2)
  More technical is Chris Harrow's look at standard deviations and sample size for his secondary math classroom:
https://casmusings.wordpress.com/2016/02/16/straightening-standard-deviations/

3)
  Meanwhile, Deborah Mayo questions Nate Silver's explication of confidence intervals (includes interesting comment thread):
http://errorstatistics.com/2016/02/12/rubbing-off-uncertainty-confidence-and-nate-silver/

4)  And just this morning comes a long post from Jim Propp on probabilities involved in lottery values (and hat detection ;-):
https://mathenchant.wordpress.com/2016/02/16/when-not-to-expect-what-youre-expecting-3/

If stats aren't your thing then skip all these, but really, everyone should be acquainting themselves more with statistics.


Monday, February 15, 2016

Infer THIS!


A fair amount to chew on in this entry from Michael Harris, focusing on the quirky (Harris says, "eclectic") online journal Inference (current issue includes an interesting Gregory Chaitin piece, as well as the set/category theoretical piece Harris concentrates on):

https://mathematicswithoutapologies.wordpress.com/2016/02/14/intelligent-design-via-higher-categories/

 ...though, if you're philosophy-phobic, perhaps best to avoid :-[
(Harris also mentions the loose theory some have that controversial writer David Berlinski is behind, or at least associated with, the journal, whose editors/progenitors go unnamed.)


Sunday, February 14, 2016

Perspective....


From whimsical Lewis Carroll, a rather different Sunday reflection:
"I may as well just tell you a few of the things I like, and then, whenever you want to give me a birthday present (my birthday comes once every seven years, on the fifth Tuesday in April) you will know what to give me. Well, I like, very much indeed, a little mustard with a bit of beef spread thinly under it; and I like brown sugar — only it should have some apple pudding mixed with it to keep it from being too sweet; but perhaps what I like best of all is salt, with some soup poured over it. The use of the soup is to hinder the salt from being too dry; and it helps to melt it. Then there are other things I like; for instance, pins — only they should always have a cushion put round them to keep them warm. And I like two or three handfuls of hair; only they should always have a little girl’s head beneath them to grow on, or else whenever you open the door they get blown all over the room, and then they get lost, you know."

Thursday, February 11, 2016

Anyone Who'd Visit a Psychiatrist Ought to Have His Head Examined


Fun with self-reference....

I've long thought that "self-reference" and "recursion" are among the most important topics out there (and I'm in good company, since they've been central to a lot of Douglas Hofstadter's work ;-). They straddle, in crucial ways, the fields of linguistics/cognitive science and math/logic. So I was delighted this week to see Ben Orlin do his humorous take on self-reference (and even Gödel's Incompleteness Theorem) and chop it down to size:

http://mathwithbaddrawings.com/2016/02/10/faqs-about-self-reference/comment-page-1/

So much fun! ...but Ben is not the only funny one... surfing around, I found this page of self-reference jokes/humor:
http://web.maths.unsw.edu.au/~jim/selfref.html

And Alexander Bogomolny also has an entertaining page of examples:
http://www.cut-the-knot.org/selfreference/

...and then there's always this xkcd classic:


...Lastly, always remember that, "All generalizations are false, including this one" (Mark Twain).


Wednesday, February 10, 2016

Not even sure what to title this!!?


Wonderful dot patterns....

Wow!. Yesterday, Patrick Honner tweeted out a link to this fabulous, freaky Numberphile video from December that I missed. Of course, all the Numberphile clips are great, but this is already one of my favorites (Patrick called it "mind-blowing")... seems like it perhaps crosses the boundaries of several mathematical/cognitive fields:




Monday, February 8, 2016

Bang?


OK, we'll kickstart the week with a recent problem from the DataGenetics blog... a fairly simple, straightforward (...and morbid ;-) probability conundrum, I've re-written below:

I say "morbid" because it involves playing "Russian roulette" with the following twist... after duct-taping you to an IKEA chair a villain pulls out an empty 6-chamber gun and loads it (in your full view) with just two bullets in ADJACENT chambers. He closes the gun cylinder, gives it a good spin, points the barrel at your carefully-coiffed head, and pulls the trigger. CLIIIICK... no bullet... maybe you've lived to enjoy another day... or... NOT. 
The villain announces he is going to pull the trigger one more time, and if you're still alive you're then free to go merrily home, or to the nearest brew pub. He'll even give you a choice: he can pull the trigger again right where he left off, or he can re-spin the cylinder again first. Which should you have him do?
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answer:   pull the trigger without doing a re-spin... 75% chance of survival in that case (3 of 4 possible remaining chambers being blank)  vs. 2/3 chance of survival with a re-spin (4 out of 6 chambers empty) .  If that's not clear, follow link back to DataGenetics for fuller explanation; and they also worked out several further variations worth checking out.


Sunday, February 7, 2016

Regression to the Mean


For today's Sunday reflection, this passage from a journal reviewer quoted in Gary Smith's "Standard Deviations":
"There are few statistical facts more interesting than regression to the mean for two reasons. First, people encounter it almost every day of their lives. Second, almost nobody understands it.
"The coupling of these two reasons makes regression to the mean one of the most fundamental sources of error in human judgment, producing fallacious reasoning in medicine, education, government, and, yes, even sports."
And speaking of statistics, this ;-):
"The average human has about one breast and one testicle."
-- Statistics 101

Wednesday, February 3, 2016

Strip Tease...


The Möbius strip of course...

I suspect we all think we've read enough about Möbius strips in the last many years (decades?)... but I'd encourage folks to make time for Evelyn Lamb's current read on the topic (one of her "favorite spaces"):

http://blogs.scientificamerican.com/roots-of-unity/a-few-of-my-favorite-spaces-the-moebius-strip/

She relates it back to the four-color-theorem, which is not often done (or six-color-theorem in this case). And these days, if you're expounding on Möbius strips, it's almost obligatory to include Vi Hart's story based on the same. I'm happy to report that Dr. Lamb does so at the end.


Tuesday, February 2, 2016

A New Find (for me)


I've indicated before my love for flea markets and thrift stores. And it's always a good day when I find some math book gem at such a site for a buck or less.
...Today, was a good day!

At a local thrift I stumbled upon a volume I'd not seen before from 1987. It's from Oxford University Press (no slouch of a publisher ;-) so I suspect many of you will be familiar with it: "Discovering Mathematics: The Art of Investigation" by British writer A. Gardiner.**  Dover continues to publish it, so it is still readily available today: http://amzn.to/1UIbmn1

I'm not all that far in yet, but my first impression is that it appears very interesting, engaging, and a fun treatment even though in a slightly textbook-ish format... and, perhaps in line with what a lot of current math education reform is attempting to accomplish.

The author's focus is on "mathematical discovery," similar to what others would call "mathematical thinking."

From the back cover:
"The word 'mathematics' usually conjures up a world of more-or-less familiar problems to be solved by more-or-less familiar techniques. This book examines a very different aspect of mathematics, namely how one can begin to explore unfamiliar, fresh ideas and chance observations, how one can pursue them through various stages until the light eventually begins to dawn, and how this whole process invariably throws up other interesting questions one would otherwise never have thought of."

Anyway, take this as a recommendation... if I change my mind as I get further into it, I'll update this post.

Here's an earlier review from MAA:
http://www.maa.org/press/maa-reviews/discovering-mathematics-the-art-of-investigation

**  I've seen the author listed variously as "Anthony Gardiner" and "Alan Gardiner;" if someone knows for sure the correct designation I'd appreciate the information.