"New ‘Social Justice’ Math Class Teaches Kids That Math Is Evil, Dehumanizing"

Via Twitter:  Millions of K-12 students across the country believe that mathematics is a sadistic discipline—(I should know, I was one of them)—but a new "social justice" training module aims to persuade teachers that maybe the kids are on to something.

The course was designed by Teach for America and is offered through EdX, according to Campus Reform. It presupposes that math could be made more interesting for students if it was infused with socially relevant themes. That's not a terrible assumption—maybe young people would like math better if it was being taught in a language they understood. (If Olivia eats 10 pieces of avocado toast every day, how long will it be until she can afford to move out of her parent's house? That sort of thing.)

But Teach for America thinks that language is "social justice," and has designed a course that makes some startling claims about math.

"In western mathematics, our ways of knowing include formalized reasoning or proof, decontextualization, and algorithmic thinking, leaving little room for those having non-western mathematical skills and thinking processes," the training course claims.

It continues:

"Mathematical ethics recognizes that, for centuries, mathematics has been used as a dehumanizing tool… mathematics formulae also differentiate between the classifications of a war or a genocide and have been used to trick indigenous peoples out of land and property."

Math is such a basic building block that one can cherry-pick hundreds of examples of it being misapplied for nefarious ends—but that's not really math's fault. Math lacks—to borrow a social justice term—agency.

I'm open to the idea that math—particularly advanced math—is over-valued as a K-12 subject. There's a good argument to be made that high schoolers should be taking less Algebra II and reading more Shakespeare. But if we're going to teach math, I'm not sure we should be teaching that it's mostly just this bad thing Western countries used to subjugate indigenous peoples, as if that's the main thing you need to know about math.

Math problem... with no right or wrong answer

Woman's March births a tweet...

Puzzlement ensues...

Grace Hopper explains Nanoseconds.

Link to video

"It's "astonishingly likely" humanity is not unique."

According to astronomer Adam Frank from the University of Rochester, one of the problems with the Drake equation is that it incorporates the hypothetical length of time advanced civilisations exist for – something we're perhaps not well equipped to be speculating about.

"The fact that humans have had rudimentary technology for roughly 10,000 years doesn't really tell us if other societies would last that long or perhaps much longer," says Frank.

But by reformulating the equation to look at the history of the whole Universe instead, the team argues that they can avoid the ambiguity of longevity estimates.

"Rather than asking how many civilisations may exist now, we ask 'Are we the only technological species that has ever arisen?'" said fellow researcher Woodruff Sullivan from the University of Washington. "This shifted focus eliminates the uncertainty of the civilisation lifetime question and allows us to address what we call the 'cosmic archaeological question' – how often in the history of the universe has life evolved to an advanced state?" (read more)

Related: Universe Likely Has Many Extinct Civilizations  

The biggest yet found prime number

New Scientist: It’s time for a new prime to shine. The largest known prime number is now 274,207,281 – 1, smashing the previous record by nearly 5 million digits.

This mathematical monster was discovered by Curtis Cooper at the University of Central Missouri in Warrensburg as part of the Great Internet Mersenne Prime Search (GIMPS), a collaborative effort to find new primes by pooling computing power online. It has 22,338,618 digits in total.

The GIMPS software automatically crunches through numbers, testing whether they are prime – that is, only divisible by themselves and one...

The prime numbers are infinite, and there is little practical use in discovering one, but the search is a good way to put computing hardware through its paces. Unrelated to the discovery of the new prime, GIMPS recently helped discover a bug in Intel’s new Skylake processors, which were crashing under the heavy workload.

"Why Pi Matters"

"Pi does deserve a celebration, but for reasons that are rarely mentioned. In high school, we all learned that pi is about circles. Pi is the ratio of a circle’s circumference (the distance around the circle, represented by the letter C) to its diameter (the distance across the circle at its widest point, represented by the letter d). That ratio, which is about 3.14, also appears in the formula for the area inside the circle, A = πr2, where π is the Greek letter “pi” and r is the circle’s radius (the distance from center to rim). We memorized these and similar formulas for the S.A.T.s and then never again used them, unless we happened to go into a technical field, or until our own kids took geometry."

 

"So it’s fair to ask: Why do mathematicians care so much about pi? Is it some kind of weird circle fixation? Hardly. The beauty of pi, in part, is that it puts infinity within reach. Even young children get this. The digits of pi never end and never show a pattern. They go on forever, seemingly at random—except that they can’t possibly be random, because they embody the order inherent in a perfect circle. This tension between order and randomness is one of the most tantalizing aspects of pi." (read the whole thing)

'The Pursuit of Beauty: Bounded Gaps Between Primes'

"No formula predicts the occurrence of primes—they behave as if they appear randomly. Euclid proved, in 300 B.C., that there is an infinite number of primes. If you imagine a line of all the numbers there are, with ordinary numbers in green and prime numbers in red, there are many red numbers at the beginning of the line: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47 are the primes below fifty. There are twenty-five primes between one and a hundred; 168 between one and a thousand; and 78,498 between one and a million. As the primes get larger, they grow scarcer and the distances between them, the gaps, grow wider."

"Prime numbers have so many novel qualities, and are so enigmatic, that mathematicians have grown fetishistic about them. Twin primes are two apart. Cousin primes are four apart, sexy primes are six apart, and neighbor primes are adjacent at some greater remove. From “Prime Curios!,” by Chris Caldwell and G. L. Honaker, Jr., I know that an absolute prime is prime regardless of how its digits are arranged: 199; 919; 991. A beastly prime has 666 in the center. The number 700666007 is a beastly palindromic prime, since it reads the same forward and backward. A circular prime is prime through all its cycles or formulations: 1193, 1931, 9311, 3119. There are Cuban primes, Cullen primes, and curved-digit primes, which have only curved numerals—0, 6, 8, and 9. A prime from which you can remove numbers and still have a prime is a deletable prime, such as 1987. An emirp is prime even when you reverse it: 389, 983. Gigantic primes have more than ten thousand digits, and holey primes have only digits with holes (0, 4, 6, 8, and 9). There are Mersenne primes; minimal primes; naughty primes, which are made mostly from zeros (naughts); ordinary primes; Pierpont primes; plateau primes, which have the same interior numbers and smaller numbers on the ends, such as 1777771; snowball primes, which are prime even if you haven’t finished writing all the digits, like 73939133; Titanic primes; Wagstaff primes; Wall-Sun-Sun primes; Wolstenholme primes; Woodall primes; and Yarborough primes, which have neither a 0 nor a 1."

Study: "Rote memorization plays crucial role in teaching students how to solve complex calculations"

"By tracking a group of young students over the course of a year, the authors show “that children learn to associate individual problems with the correct answers. Repeated problem solving during the early stages of arithmetic skill development also contributes to memory re-encoding and consolidation, thus resulting in enhanced hippocampal activity and ability to recall basic arithmetic facts… The maturation of problem-solving skills is characterized by a gradual decrease in the use of inefficient procedures such as counting and an increase in the use of memory-based strategies.”

"As a scientific justification of rote learning, the study seems likely to further polarize the controversy over math teaching styles, in which arithmetical fundamentalists are squared off against the popular and progressive forces of “discovery-based” learning, in which students are encouraged to find their own ways to the right answer." (read the whole thing)

Via Instapundit

NPR: How To Marry The Right Girl, A Mathematical Solution

"Imagine that you are interviewing 20 people to be your secretary [or your spouse or your garage mechanic] with the rule that you must decide at the end of each interview whether or not to give that applicant the job." If you offer the job to somebody, game's up. You can't go on and meet the others. "If you haven't chosen anyone by the time you see the last candidate, you must offer the job to her," Alex writes (not assuming that all secretaries are female — he's just adapting the attitudes of the early '60s).

So remember: At the end of each interview, you either make an offer or you move on.

If you don't make an offer, no going back. Once you make an offer, the game stops.

According to Martin Gardner, who in 1960 described the formula (partly worked out earlier by others), the best way to proceed is to interview (or date) the first 36.8 percent of the candidates. Don't hire (or marry) any of them, but as soon as you meet a candidate who's better than the best of that first group — that's the one you choose! Yes, the Very Best Candidate might show up in that first 36.8 percent — in which case you'll be stuck with second best, but still, if you like favorable odds, this is the best way to go.

Why 36.8 percent? The answer involves a number mathematicians call "e" – which, reduced to a fraction 1/e = 0.368 or 36.8 percent. For the specific details, check here, or Alex's book, but apparently this formula has proved itself over and over in all kinds of controlled situations. While it doesn't guarantee happiness or satisfaction, it does give you a 36.8 percent chance — which, in a field of 11 possible wives — is a pretty good success rate.

Also see Freeman Hunt's "Frank Advice for a Male Relative on Finding a Mate" Good Luck... and don't call me Frank ;)

Probability Pundit: Processing the 2016 GOP Primaries

What is Quantum Mechanical Superposition?

Scientific American: "Equations Are Art inside a Mathematician’s Brain"

"When mathematicians describe equations as beautiful, they are not lying. Brain scans show that their minds respond to beautiful equations in the same way other people respond to great paintings or masterful music. The finding could bring neuroscientists closer to understanding the neural basis of beauty, a concept that is surprisingly hard to define."

In the study, researchers led by Semir Zeki of University College London asked 16 mathematicians to rate 60 equations on a scale ranging from "ugly" to "beautiful." Two weeks later, the mathematicians viewed the same equations and rated them again while lying inside a functional magnetic resonance imaging (fMRI) scanner. The scientists found that the more beautiful an equation was to the mathematician, the more activity his or her brain showed in an area called the A1 field of the medial orbitofrontal cortex.

Skipping down to the last three paragraphs...

The study found, for example, that the beauty of equations is not entirely subjective. Most of the mathematicians agreed on which equations were beautiful and which were ugly, with Euler's identity, 1+eiπ=0, consistently rated the most attractive equation in the lot. "Here are these three fundamental numbers, e, pi and i," Adams says, "all defined independently and all critically important in their own way, and suddenly you have this relationship between them encompassed in this equation that has a grand total of seven symbols in it? It is dumbfounding."

On the bottom of the heap, mathematicians consistently rated Srinivasa Ramanujan's infinite series for 1/π most ugly.

"It doesn't sing," Adams says. "I look at it, and I don't learn anything new about pi. And those numbers, 26,390? 9801? As far as I am concerned, you could switch in other numbers, and I couldn't tell the difference."

Scientific American

"Math Explains Likely Long Shots, Miracles and Winning the Lottery [Excerpt]"

I'm only including the birthday problem here. For the seemingly long shot, like the lotteries, click here.

The birthday problem poses the following question: How many people must be in a room to make it more likely than not that two of them share the same birthday?

The answer is just 23. If there are 23 or more people in the room, then it's more likely than not that two will have the same birthday.

Now, if you haven't encountered the birthday problem before, this might strike you as surprising. Twenty-three might sound far too small a number. Perhaps you reasoned as follows: There's only a one-in-365 chance that any particular other person will have the same birthday as me. So there's a 364/365 chance that any particular person will have a different birthday from me. If there are n people in the room, with each of the other n − 1 having a probability of 364/365 of having a different birthday from me, then the probability that all n − 1 have a different birthday from me is 364/365 × 364/365 × 364/365 × 364/365 … × 364/365, with 364/365 multiplied together n − 1 times. If n is 23, this is 0.94.

Because that's the probability that none of them share my birthday, the probability that at least one of them has the same birthday as me is just 1 − 0.94. (This follows by reasoning that either someone has the same birthday as me or that no one has the same birthday as me, so the probabilities of these two events must add up to 1.) Now, 1 − 0.94 = 0.06. That's very small.

Yet this is the wrong calculation to consider because that probability—the probability that someone has the same birthday as you—is not what the question asked. It asked about the probability that any two people in the same room have the same birthday as each other. This includes the probability that one of the others has the same birthday as you, which is what I calculated above, but it also includes the probability that two or more of the other people share the same birthday, different from yours.

This is where the combinations kick in. Whereas there are only n − 1 people who might share the same birthday as you, there are a total of n × (n − 1)/2 pairs of people in the room. This number of pairs grows rapidly as n gets larger. When n equals 23, it's 253, which is more than 10 times as large as n − 1 = 22. That is, if there are 23 people in the room, there are 253 possible pairs of people but only 22 pairs that include you.

So let's look at the probability that none of the 23 people in the room share the same birthday. For two people, the probability that the second person doesn't have the same birthday as the first is 364/365. Then the probability that those two are different and that a third doesn't share the same birthday as either of them is 364/365 × 363/365. Likewise, the probability that those three have different birthdays and that the fourth does not share the same birthday as any of those first three is 364/365 × 363/365 × 362/365. Continuing like this, the probability that none of the 23 people share the same birthday is 364/365 × 363/365 × 362/365 × 361/365 … × 343/365.

This equals 0.49. Because the probability that none of the 23 people share the same birthday is 0.49, the probability that some of them share the same birthday is just 1 − 0.49, or 0.51, which is greater than half.

Scientific American via Instapundit 

The Kiss Precise

The Kiss Precise by Frederick Soddy

For pairs of lips to kiss maybe
Involves no trigonometry.
'Tis not so when four circles kiss
Each one the other three.
To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.

Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance from the center.
Though their intrigue left Euclid dumb
There's now no need for rule of thumb.
Since zero bend's a dead straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.

To spy out spherical affairs
An oscular surveyor
Might find the task laborious,
The sphere is much the gayer,
And now besides the pair of pairs
A fifth sphere in the kissing shares.
Yet, signs and zero as before,
For each to kiss the other four
The square of the sum of all five bends
Is thrice the sum of their squares.

Published in Nature, June 20, 1936

More Math, Less Fighting #3

This one is also from The Moscow Puzzles and expands on the idea of yesterday's problem.

White and Black 

Take 4 black and 4 white checkers (or 4 pennies and 4 other coins) and put them on a table in a row, white, black, white, black, and so on. Leave a vacant place at one end which can hold 2 checkers. After 4 moves, all the black checkers should be on one side and the white ones on the other. 

A move consists of shifting 2 adjacent checkers, keeping their order, into any vacant space.

Solution from the book to appear in the comments tomorrow.

More Math, Less Fighting #2

Here's one from the beginning of The Moscow Puzzles by Boris A. Kordemsky.

Moving Checkers 

Place 6 checkers on a table in a row, alternating them black, white, black, white, [black, white.] 

Leave a vacant place large enough for 4 checkers on the left. 

Move the checkers so that all the white ones will end on the left, followed by all the black ones. The checkers must be moved in pairs, taking 2 adjacent checkers at a time, without disturbing their order, and sliding them to a vacant place. To solve this problem, only three such moves are necessary. ... 

If no checkers are available, use coins, or cut pieces out of paper or cardboard.

Or use paperclips or post-it notes or whatever. Solution from the book to appear in the comments of this post tomorrow.