Did Someone Mention Algebra?

Predictably, the NY Times piece, "Is Algebra Necessary," cited in the prior post, has drawn a host of quick critiques. A sampling here:

http://blogs.scientificamerican.com/observations/2012/07/30/abandoning-algebra-is-not-the-answer/

http://dropoutnation.net/2012/07/30/why-algebra-matters-and-why-those-who-think-it-doesnt-are-wrong/

http://scientopia.org/blogs/galacticinteractions/2012/07/29/when-andrew-hacker-asks-is-algebra-necessary-why-doesnt-he-just-ask-is-high-school-necessary/

http://www.washingtonpost.com/blogs/answer-sheet/post/yes-algebra-is-necessary/2012/07/30/gJQAr6xMKX_blog.html

http://soffer801.wordpress.com/2012/07/30/algebra-is-necessary/

and, finally, Mark Chu-Carroll, as usual, having difficulty expressing his real thoughts ;-) :

http://scientopia.org/blogs/goodmath/2012/07/30/mathematical-illiteracy-in-the-nyt/

Algebra... Who Needs It!

(pic via Wikimedia Commons)

The NY Times adds to the fray over math education here; specifically, the usefulness of "algebra":

http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html?_r=1&smid=tw-share

Some excerpts therefrom: 

"There are many defenses of algebra and the virtue of learning it. Most of them sound reasonable on first hearing; many of them I once accepted. But the more I examine them, the clearer it seems that they are largely or wholly wrong — unsupported by research or evidence, or based on wishful logic."

"Making mathematics mandatory prevents us from discovering and developing young talent. In the interest of maintaining rigor, we’re actually depleting our pool of brainpower. I say this as a writer and social scientist whose work relies heavily on the use of numbers. My aim is not to spare students from a difficult subject, but to call attention to the real problems we are causing by misdirecting precious resources."

"Nor is it clear that the math we learn in the classroom has any relation to the quantitative reasoning we need on the job."

"Instead of investing so much of our academic energy in a subject that blocks further attainment for much of our population, I propose that we start thinking about alternatives. Thus mathematics teachers at every level could create exciting courses in what I call 'citizen statistics'... it would familiarize students with the kinds of numbers that describe and delineate our personal and public lives."

"The aim would be to treat mathematics as a liberal art, making it as accessible and welcoming as sculpture or ballet. If we rethink how the discipline is conceived, word will get around and math enrollments are bound to rise. It can only help."

I don't agree with much of the article, but if it's in the Times it will get some conversation going (I suspect one could generate some similarly-framed arguments on why taking 4 years of high school English is a waste of students' time, let alone reading Shakespeare or other literature "classics"); still, the underlying impetus for reforming math education, to one degree or another, will garner much agreement.

Indeed, "MathBabe" quickly responded to the article at her own blog (and is drawing a lot of further comments) here:

http://mathbabe.org/2012/07/29/does-mathematics-have-a-place-in-higher-education/ 

Towards the end she writes thusly:

"Finally, I’d say this (and I’m stealing this from my friend Kiri, a principal of a high school for girls in math and science): nobody ever brags about not knowing how to read, but people brag all the time about not knowing how to do math. There’s nothing to be proud of in that, and it’s happening to a large degree because of our culture, not intelligence.

So no, let’s not remove mathematical literacy as a requirement for college graduates, but let’s think about what we can do to make the path reasonable and relevant while staying rigorous."

 Amen to that....

Come Friday Haz Puzzle

An older, "simple self-referential test" from Patrick Honner today (presented in duplicate for some reason). Just 5 questions, so don't sue me if it ties your brain fissures in knots:

http://tinyurl.com/ce2dv36

answer below:
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ANSWER:

B C C D A (…not sure if there's more than one answer set that works)

Air Flight Networking & Epidemics

Where math, germs, and airplanes intersect:

http://www.npr.org/blogs/health/2012/07/24/157306307/epidemics-prefer-changing-planes-in-jfk-over-atl?ft=1&f=1007

A piece about which airports are likely the greatest hubs in the quick spread of epidemics.

Quote… Unquote

Yet again, I'm pulled back to prime numbers for a post… GREAT compendium of quotes pertaining to prime numbers, the zeta function, and Riemann Hypothesis here:

http://empslocal.ex.ac.uk/people/staff/mrwatkin/isoc/quotes.htm

The page comes from (Brit) Matthew Watkins, who co-authored "The Mystery of the Prime Numbers," a volume I haven't read myself, but have seen several uniformly positive reviews of. His webpage linking to prime number-related stuff is here:

http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics.htm

And here a small sampling of quotes from the initially-cited page:

"How can so much of the formal and systematic edifice of mathematics, the science of pattern and rule and order per se, rest on such a patternless, unruly, and disorderly foundation? Or how can numbers regulate so many aspects of our physical world and let us predict some of them when they themselves are so unpredictable and appear to be governed by nothing but chance?"
   -- H. Peter Aleff, from the 'e-book' Prime Passages to Paradise
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"We may - paraphrasing the famous sentence of George Orwell - say that "all mathematics is beautiful, yet some is more beautiful than the other." But the most beautiful in all mathematics is the zeta function. There is no doubt about it."   -- Polish cosmologist Krzysztof Maslanka
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"In [his 1859 paper], Riemann made an incidental remark - a guess, a hypothesis. What he tossed out to the assembled mathematicians that day has proven to be almost cruelly compelling to countless scholars in the ensuing years...

...it is that incidental remark - the Riemann Hypothesis - that is the truly astonishing legacy of his 1859 paper. Because Riemann was able to see beyond the pattern of the primes to discern traces of something mysterious and mathematically elegant at work - subtle variations in the distribution of those prime numbers. Brilliant for its clarity, astounding for its potential consequences, the Hypothesis took on enormous importance in mathematics. Indeed, the successful solution to this puzzle would herald a revolution in prime number theory. Proving or disproving it became the greatest challenge of the age...

It has become clear that the Riemann Hypothesis, whose resolution seems to hang tantalizingly just beyond our grasp holds the key to a variety of scientific and mathematical investigations. The making and breaking of modern codes, which depend on the properties of the prime numbers, have roots in the Hypothesis. In a series of extraordinary developments during the 1970s, it emerged that even the physics of the atomic nucleus is connected in ways not yet fully understood to this strange conundrum. ...Hunting down the solution to the Riemann Hypothesis has become an obsession for many - the veritable 'great white whale' of mathematical research. Yet despite determined efforts by generations of mathematicians, the Riemann Hypothesis defies resolution."  -- J. Derbyshire, from the dustjacket description of Prime Obsession (John Henry Press, 2003)
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 "For many mathematicians working on it, $1m is less important than the satisfaction that would come from finding a proof. Throughout my researches among the mathematicians' tribe (I have interviewed 30 in the past year), Riemann's Hypotheis was often described to me in awed terms. Hugh Montgomery of the University of Michigan said this was the proof for which a mathematician might sell his soul. Henryk Iwaniec, a Polish-American mathematician, sounded as if he were already discussing terms with Lucifer"

'I would trade everything I know in mathematics for the proof of the Riemann Hypothesis. It's gorgeous stuff. I'm only worried that I'll be unable to understand it. That would be the worst...'"   -- K. Sabbagh, "Beautiful Mathematics", Prospect, January 2002
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"Proving the Riemann hypothesis won't end the story. It will prompt a sequence of even harder, more penetrating questions. Why do the primes achieve such a delicate balance between randomness and order? And if their patterns do encode the behaviour of quantum chaotic systems, what other jewels will we uncover when we dig deeper?

Those who believe mathematics holds the key to the Universe might do well to ponder a question that goes back to the ancients: What secrets are locked within the primes?"   -- E. Klarreich, "Prime Time" (New Scientist, 11/11/00)

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 There now exist a great many volumes related to prime numbers for lay folks (it's been a hot topic in recent years). Among my own favorites are (in no particular order):

"The Music of the Primes" -- Marcus du Sautoy
"The Riemann Hypothesis" -- Karl Sabbagh
"Stalking the Riemann Hypothesis" -- Dan Rockmore
"Prime Numbers" by David Wells, a more encyclopedic/academic (but still interesting) offering.

And finally, John Derbyshire's "Prime Obsession" was another hugely popular volume on the subject, though I didn't enjoy it quite as much as the preceding titles.

Not the Usual Prisoners' Dilemma

This one may make your head hurt... but still interesting if you get all the way through it (from Scientific American, another brainteaser based on prisoner choices):

http://blogs.scientificamerican.com/observations/2012/07/19/puzzling-prisoners/

Another Prime Example....

Recently came across a blurb regarding factorials in the Appendix of Laurie Buxton's old book, "Mathematics For Everyone" which I thought was quite lovely and I didn't recall reading before. I'll just quote verbatim (I've added bold):

"Now consider the string of consecutive numbers:

600! + 2
600! + 3
600! + 4
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600! + 600

There are 599 consecutive numbers here, and none of them can be prime, for every one divides by at least the number on the right. For instance, 600! + 59 must divide by 59, since every number up to 600 divides into 600! This gives us a gap of 599 with no primes. We chose 600 at random to start. It is clear that we could start with a number as large as we like, and hence make the gap as large as we like."

To me, another simple yet beautiful mathematical thought process here... in fact, makes me wonder how much factorials have been used in approaching the Riemann Hypothesis, or more generally, the distribution of primes?

Friday Puzzle

Lots of great, yet simple, card puzzles out there. For a Friday puzzle I'll just re-word one from the 'Math Is Fun' site:

10 cards are placed FACING UP into a normal 52-card deck (42 cards left facing down). The deck is shuffled several times and then handed to a blind man. He is asked to now divide the deck into 2 piles (of any sizes) such that each pile have the same number of cards facing up.
How can he do it?

For the answer, shuffle over to:

http://www.mathsisfun.com/puzzles/cards-facing-up-solution.html

Finding Primes

A 2008 post on generating prime numbers that includes lots of good links:

http://recursed.blogspot.com/2008/07/rutgers-graduate-student-finds-new.html

Tanton's Principles

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

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

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

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

10 + 10 …to Enjoy

This blog is aimed primarily (though not exclusively) at laypeople who are interested in math, but aren't math professionals, and the links provided in the right-hand column may sometimes be daunting to such folks, with sooooo many to look through.
So, since it's always fun to compile Top 10 lists anyway, I've picked out a condensed list of just 10 favorite math links that offer a range and variety of content (obviously biased by my own interests) for what I call 'math buffs.'
So without further adieu, and in no particular order:

1 general content site (Wolfram MathWorld):

http://mathworld.wolfram.com/

1 magazine style site:

http://plus.maths.org/content/

4 math blogs:

http://wildaboutmath.com/

http://www.mathteacherctk.com/blog/

http://profkeithdevlin.org/

http://mathrising.com/

2 sites with personal videos:

http://www.jamestanton.com/?category_name=curriculum-tidbits

http://www.youtube.com/user/njwildberger/videos

2 sites that link to further math-related videos:

http://www.cosmolearning.com/mathematics/documentaries/
(documentaries largely from the BBC)

http://pinterest.com/mathematicsprof/

Additionally, I can't help but mention that Wikipedia's portal site for mathematics, links to a plethora of further math content:

http://en.wikipedia.org/wiki/Portal:Mathematics

I'm leaving out a bazillion great math sites in citing the above 10+… in fact I started to do a list of honorable mentions, but there were sooooo many I wanted to include, I simply jettisoned the idea!! But hope you enjoy some of the above choices.

And here, for those of you on Twitter, 10 of my favorite math-oriented tweeters, in simple alphabetical order (some overlap of course with the above selections, and again, I've left out an awful LOT of wonderful twitterers in holding the list to just 10). Check 'em out if you don't already follow them:

https://twitter.com/CutTheKnotMath 
https://twitter.com/grey_matter 
https://twitter.com/JamesTanton 
https://twitter.com/MathematicsProf
https://twitter.com/MatthewMaddux
https://twitter.com/MrHonner
https://twitter.com/ProfKeithDevlin
https://twitter.com/RepublicOfMath
https://twitter.com/StevenStrogatz 
https://twitter.com/WilderLab

....wish I coulda grown up when there was SO MUCH great math content at one's fingertips!!

Friday puzzle

A state basketball tournament starts off with 25 teams. Two teams play at a time with the losing team eliminated from the tournament (single-elimination contest). How many games must be played to crown the eventual champion?

Answer below:
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ANSWER:

24 games

A Passion For Primes

If you don't think prime numbers can be entertaining, watch this TEDTalk:

NJ Wildberger Videos

More instructional math videos on the Web that look interesting:

http://www.youtube.com/user/njwildberger/videos

(brought to my attention by 'Republic of Math')

Just Asketh...

Got a math question you want answered? Murray Bourne over at SquareCircleZ put together this list (which his commenters have added to) of digital resources for getting answers:

http://www.intmath.com/blog/top-10-math-help-sites/5966

Counting With One's Fingers…

...isn't just child's play… new research indicates that the method one uses for counting using the hand and fingers has cognitive consequences, having to do with embodied cognition, the idea that cognition is distributed throughout parts of the body and not limited to the brain:

http://www.guardian.co.uk/science/blog/2012/jun/26/count-fingers-brain

 from the piece:

"...the extent of cultural diversity in finger-counting has been hugely underestimated... by studying finger counting techniques, we could better understand how culture influences cognitive processes – particularly mental arithmetic.
"There is a mental link between hands and numbers, but that link doesn't come from humans learning to use their hands as a counting aid. It goes back much further in our evolution...

"[finger-counting] affects how we mentally represent and process numbers. That may be because finger counting has one unique property that sets it apart from written or verbal counting systems: it is a sensory-motor experience, with a direct link between bodily movement and brain activity…."

Friday Puzzle

Simple logic conundrum for today...

Before you on a table are 3 closed boxes, one full of RED chips, one full of BLUE chips, and one with a mix (half-and-half) of RED and BLUE chips. The boxes are labeled "RED," "BLUE," and "Mixed," but each label is WRONG (on the incorrect box). You are to blindly and randomly pick ONE chip out of any box of your choice and then re-assign the labels correctly (upon seeing the picked chip). Can it be done, and if so from which box would you select?

ANSWER below:
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pick from box labeled "Mixed" -- whatever you get, red or blue, will be proper label for that box (since it can't actually be mixed), and then the other two follow easily.

Khan & Its Critics

Important (and long) post from Keith Devlin on the skirmishes between Khan Academy and its critics here (includes many good links as well):

http://devlinsangle.blogspot.com/2012/07/cant-we-all-get-along.html

...should be read by everyone interested in math education (...but then that probably goes for everything Keith writes!)

excerpt:

"Learning mathematics is hard. Very hard. It is easy to get discouraged and give up. Some of us, when we are learning math the first time, are lucky enough to have a parent, grandparent, uncle or aunt, older sibling, or family friend who can sit down alongside us and help us. I suspect that a great many of today’s professional mathematicians owe their eventual success in the subject to someone who mentored them in the early days.

"But not everyone has such a person in their lives. At least, they did not until Sal Khan came along: friendly, non-threatening, patient, and a good explainer (actually not brilliant, but that might be all to the good, since a brilliant instructor could easily discourage a less-brilliant student). Above all, human. A regular guy. Just think about that for a moment. It’s a valuable weapon in the educational landscape....

"...Then Bill Gates comes along, and KA goes global. Expectations change. Now things have gotten more tricky. When a resource like KA becomes the primary vehicle by which millions of people acquire many or all of their mathematical skills, the stakes become dramatically higher than when it was a one-man homework-help service. Like it or not, ask for it or not, KA now has (in my view) an obligation to get things right. Doing so without destroying a major part of its appeal, is clearly going to be tricky."

Video Portal

A Pinterest portal to mathematics videos created here (from @MathematicsProf):

http://pinterest.com/mathematicsprof/

...looks like a place any math buff could definitely wile away some time!

It's Summertime... It's Hot!

...It's T-Shirt time!!

Stock up from my Zazzle offerings for the blazing months ahead (buy 1 or 101).

Eight of the 30 available selections from:

http://www.zazzle.com/SheckyR