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? Read More »
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.”) Read More »
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.
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. Read More »
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. Read More »
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. Read More »
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. Read More »