Search the Site

Posts Tagged ‘Mathematics’

Street-Fighting Math MOOC Is Open

My mathematically inclined readers are cordially invited to enroll in “6.SFMx: Street-Fighting Math,” which starts today on EdX. Like most (all?) MOOC courses, it is free and open to world, as are all the course materials.

So far, I have learned that teaching an entirely online course requires far more effort than teaching in person. Maybe by a factor of 10. Partly, it is the difference between talking to a friend on the phone—you just pick up the phone and start talking—compared to writing a long letter that needs to be thought out. To this difference you add that 10,000 others will also read and depend on the letter. You get nervous about making all the pieces right. They never will be, so you never rest easy.


Book Review: Malcolm Gladwell's David and Goliath

Malcolm Gladwell’s latest — David and Goliath: Underdogs, Misfits, and the Art of Battling Giants – came out this week.   Like every other book by Gladwell, it is already a best-seller. And having read – and very much enjoyed — the book, I can see why. Gladwell once again presents a variety of interesting stories, this time centered on the question of whether underdogs are as disadvantaged as we believe (the opening story on David and Goliath – which makes this observation – is worth the price of admission).   My sense – from the few reviews I have seen – is that critics have primarily focused on whether the argument they think Gladwell is making is valid.  I am going to argue that this approach misses the fact that the stories Gladwell tells are simply well worth reading (i.e., these stories are interesting and make you think). 

The range of stories Gladwell presents is quite impressive. From the opening biblical story to a discussion of the number of students in a school classroom, the impact of dyslexia, the curing of leukemia, the battle for Civil Rights, French revolutionaries during World War II, etc… One has to wonder: where does Gladwell find these stories?


Beware the Weasel Word "Statistical" in Statistical Significance!

As Justin Wolfers pointed out in his post on income inequality last week, the Census Bureau was talking statistical nonsense. I blame the whole idea of statistical significance. For its weasel adjective “statistical” concedes that the significance might not be the kind about which you care. Here, I’ll explain what statistical significance is, and how its use is harmful to society.

To evaluate the statistical significance of an effect, you calculate the so-called p value; if the p value is small enough, the effect is declared statistically significant. For an example to illustrate the calculations, imagine that your two children Alice and Bob play 30 rounds of the card game “War,” and that the results are 20-10 in favor of Bob. Was he cheating?

To calculate the p value, you need an assumption, called the null (or no-effect) hypothesis: here, that the game results are due to chance (i.e. no cheating). The p value is the probability of getting results at least as extreme as the actual results of 20-10. Here, the probability of Bob’s winning at least 20 games is 0.049. (Try it out at Daniel Sloper’s “Cumulative Binomial Probability Calculator.”)


Division, Not Long Division

Division is the most powerful arithmetic operation. It makes comparisons. When the numerator and denominator have the same units, the comparison makes a dimensionless number, the only kind that the universe cares about. Long division, however, is something else entirely. In my post “Dump algebra,” many commentators objected to my loathing of long division. But long division is not division! Long division is just one way to do the computation, and is far from the most useful way.

I’ll illustrate with an actual example of division. For my environmental-protection lawsuit, now in the Massachusetts Supreme Court, I needed to divide 142,500 by 4655.


Dump Algebra

Being a good teacher, I like to think, requires a curious and freethinking mind. A supporting example is Andrew Hacker, described by a former Cornell colleague as “the most gifted classroom lecturer in my entire experience of 50 years of teaching.” His book Higher Education?: How Colleges Are Wasting Our Money and Failing Our Kids—and What We Can Do About It, co-authored with Claudia Dreifus, convinced me that tenure is harmful. His latest broadside, “Is Algebra Necessary?”, in last Sunday’s New York Times, is as provocative.

He argues that we should stop requiring algebra in schools. Despite the vitriol in several hundred comments (“We read them so you don’t have to.”), he is right.


Ending the Math Wars in a Treaty of QAMA

As a country, we are often at war. If it’s not against Germany, England, terrorism, or Grenada, it’s the war on poverty (that’s gone so well), the war on cancer (ditto), and, of particular interest to me, the Math Wars, which have been raging for decades. On one side, the traditionalists insist on drilling and back to basics, “on behalf of sanity and quality in math education.” On the other side, the reformers insist on conceptual understanding using computers and calculators, to “promot[e] the rational reform of mathematics education.”

Both are half-right and half-crazy. As the reformers say, students need to understand what the mathematics means. Students whose word problem for “6 x 3 = 18” is of the form “There were 6 ducks, and 3 more showed up, so 6 times 3 is 18,” understand little. (See “Children Learning Multiplication, Part 1,” in the articles by Professor Thomas C. O’Brien.) As the traditionalists say, using computers for everything leads to needing a calculator to compute what 6.5 x 10 is.

However, there’s a tool to combine the merits of both sides: the Quick, Approximate, Mental Arithmetic (QAMA) calculator.


The Mathematics of Magic

I don’t particularly like math.  I’ve never been a fan of magic either.  For some reason, however, when I heard about a new book entitled Magical Mathematics written by two first-rate mathematicians, Persi Diaconis and Ron Graham, I felt compelled to buy it and read it.

I have to say that it is really good, and I would highly recommend it to any nerd.  It is a really artful melding of card tricks that are remarkable, with explanations of the underlying math concepts that are at one level so simple and clear that almost anyone could get the basic intuition for what they are talking about, but at another level so deep and difficult that it is probably hopeless for someone like me to ever truly understand. 


Addition Is Useless, Multiplication Is King: Channeling Our Inner Logarithm

TIME magazine has been running a series called “Brilliant: The science of smart” by Annie Murphy Paul. The latest column, “Why Guessing Is Undervalued,” quoted several results from research on learning estimation, a topic near to my heart. One result surprised me particularly:

…good estimators possess a clear mental number line — one in which numbers are evenly spaced, or linear, rather than a logarithmic one in which numbers crowd closer together as they get bigger. Most schoolchildren start out with the latter understanding, shedding it as they grow more experienced with numbers.

I do agree that children start out with a logarithmic understanding. I first learned this idea from a wonderful episode of WNYC’s Radio Lab on “Innate numbers” (Nov. 30, 2009). The producers had asked Stanislas Dehaene to discuss his research on innate number perception.

One of his studies involved an Indian tribe in the Amazon. This tribe does not have words for numbers beyond five, and does not have formal teaching of arithmetic.


Healthcare Needs Insurance Companies Like a Fish Needs a Bicycle

My reveries about toasts landing butter side down — the subject of an upcoming blog entry — were pleasantly broken by the discussion on this blog of Ezekiel Emanuel’s analysis of healthcare spending, in which Emanuel goes to town on Left and Right policy ideas for cutting spending. Among his criticisms of Left ideas:

[I]t turns out that the combined profits of the country’s five largest for-profit health insurance companies — United, WellPoint, Aetna, Humana and Cigna — were $11.7 billion, only 0.5 percent of total health-care spending. Even confiscating every penny of those profits would add up to less than half of the cost-saving threshold.

I liked how he quoted the savings as a dimensionless number (the percentage of total healthcare spending).


The Way We Teach Math, Sciences, and Languages Is Wrong

A few years after I learned German, I got the chance to learn French. That experience gave me lots of ideas for why our teaching of many subjects, especially science and mathematics, is so unsuccessful—and for how we can improve our learning.

I studied French in school for five years. However, when I went to France after college, I could barely buy a train ticket. The impetus to try again came a few years later, in the summer of 1993. Our whole family was going to spend two months in Lyon while my father took a sabbatical. The rest of us enrolled in a four-week language course at the Alliance Française.

While still in America, to get more benefit from the language course, I started relearning French. On the recommendation of a friend who is a linguist and mathematician, I got the self-study French course made by Assimil entitled Le Nouveau Français sans Peine (New French With Ease). (Many other self-study courses should also work well. I have not tried them, so I do not have the knowledge to draw out lessons for learning other subjects, which is my main interest here. But to learn about language programs, I recommend the excellent “How to learn any language” site.)

I did one French lesson daily starting from Lesson 1. I read a short, idiomatic dialogue out loud using the pronunciation key, then listened to it on the tape, repeating it sentence by sentence. The lesson finished with 2 minutes of fill-in-the-word exercises using the vocabulary from the dialogue. Each lesson took about 30 minutes. After three months of this preparation, when I landed in France I could converse with random French people on the train.


How Richard Feynman Thought

I am fascinated by how we can improve our thinking and problem solving and enjoy learning about and from masters of those arts. My interest was therefore caught by the advice on thinking given in a review of Quantum Man: Richard Feynman’s Life in Science. The reviewer, George Johnson, writes:

This triumph came early in his [Feynman’s] career. His later thinking (about solid-state physics, for example, or quantum cosmology) was just as original. Maybe sometimes too original, Krauss suggests. Science usually proceeds by building on what came before. The maverick in Feynman kept him from accepting even the most established ideas until he had torn them apart and reassembled the pieces. That led to a deeper understanding, but his time might have been better spent at the cutting edge…“He continued to push physics forward as few modern scientists have,” Krauss [the biographer] writes, “but he tended to lead from the rear or, at best, from a side flank.”


When Estimating Is Dangerous

A recent article in The New York Times offers a worrying application of street-fighting reasoning methods. The article describes the deterioration of the Lake Isabella Dam in California. This dam, the article reports, is one of 4,400 considered “susceptible to failure” (out of the 85,000 dams in the country). I’ll pass over lightly the statement early in the article that the repair costs would be “billions of dollars,” and note only that this figure seems like a massive underestimate.


The Real Cost of Unnecessary Breast Biopsies

Articles on the health-care industry are a fertile source of large numbers and, sometimes, large errors. It is estimated that nationally 300,000 women a year may be getting unnecessary surgery at a cost of “hundreds of millions of dollars.” I was happy to believe the figure of 300,000 women a year. However, the cost set off my number-sense alarm.


The Rejects' Journal

As Levitt once noted, getting a paper published in an academic journal is an often comically belabored process, and of course it’s even worse when the paper is rejected.


Funny Math

| For admirers of Indexed, we bring you: New Math. On this site, Craig Damrauer offers up one new formula each Monday to describe our world. In case you were wondering: Carjacking = Can I borrow your car? – No, you can’t. [%comments]


Our Daily Bleg: What’s Been Said About Math?

Our resident quote bleggar Fred Shapiro, editor of the Yale Book of Quotations, is back with another request. If you have a bleg of your own — it needn’t have anything to do with quotations — send it along here. Photo: foundphotoslj Recently, after a Wall Street Journal article named The Yale Book of Quotations as the second-most-essential reference book . . .


Further Adventures in Bus Stop Strategy

Now, this is disappointing: three mathematicians go to the trouble to model bus waiting strategy — is it better to wait or to walk to the next forward stop? — and conclude that waiting is the best option. Why am I disappointed? Because they didn’t even consider an alternative bus waiting strategy discussed earlier on this blog: walking backward one . . .


A Little Math Puzzle to Ponder

My good friend Dave Eldan sends me interesting tidbits on a regular basis. More often than not, they are pulled from obituaries. Everyone needs a hobby, I guess. I found his latest missive very interesting. It is from an obituary by Morton White for the great philosopher and mathematician Willard Van Orman Quine. (Unless I am dreaming, I actually had . . .