Yow. I ignore the comments for weeks, and three or four interesting topics come up.
A) you may be interested in the Iowa Gambling Task experiments, one observation of which is (I'm paraphrasing here, so read some real research if you want to see what the studies really claim) that people can develop a sense of a probability from a smallish sample, _before_ they consciously recognize that they have such a sense. (Paraphrase 2: drawing from a deck of possible rewards/penalties, someone will start reacting negatively to a "bad" deck before they consciously decide it's a "bad" deck.)
B) [not actually chronological] LTG says "I am surprised any mathematician on this blog would expect children to have an intuitive grasp of probability. Adults sure don't...It is well known...that individuals tend...to treat a 99,999 to 1 chance...differently from...a 1 in 100,000 chance..."
This, and I really do not mean to offend, is because you are not a mathematician, particularly not a mathematician who has to teach non-mathematicians.
Mathematicians (especially in education) think about what is and isn't intuitive mathematics a great deal, because what _seems_ intuitive to us is clearly not to our students. And one big idea that's come out of this is the impetus to connect the language and structure of mathematics to the structures and intuition that "normal" people already have, rather than treat math as abstract and foreign.
So we talk about "number sense" and "proportional reasoning" and "intuitive probability" and part of what we mean is a non-linguistic concept of those things. The theory is that people have a sense of probability, both in the cases of extremely rare events and relatively common events, but:
a) your intuition can be misled, and a high-level, abstract argument lacks potency (e.g., because you've heard about miraculous recoveries from coma, your intuition is that is a reasonably likely event, and a doctor telling you it's a one in a million chance doesn't dissuade you.)
b) it is difficult to connect your intuition, in the many cases when it is right, to the abstract claims of probability, if they're couched in examples not relevant to your experience or in language you don't have other intuitive connections with (most people troubled by the Monty Hall problem didn't watch episode after episode of Let's Make a Deal; most people have never thought of what "1 in 100,000" means, in any concrete way, and usually don't have a reason to.)
Thus, while mathematicians cannot help but see that most adults' sense of probability is _solely_ intuitive, and unconnected to any attempts at probabilistic abstraction or language, and (therefore?) usually lacks _nuance_, we cling to the belief (well justified, really) that there is in fact an intuition there.
(I should point out that numbers are themselves language, as I'm using the phrase. I doubt that caused any confusion, but you can never be too careful on the interweb.)
C) I took Pombat's examples to just be illustrations of the way probabilities are bandied about in misleading ways, and not a claim about DNA evidence per se. For more examples of this and other grievous misquantification, John Allen Paulos' Innumeracy (and/or A Mathematician Reads the Newspaper) are excellent reading.
D) RbR's argument (and I know I shouldn't say this, but I just can't help it, so please forgive me) assumes facts not in evidence. A claim about the sensitivity of a forensic test is just that, independent of other factors. If a jury is deciding guilt or innocence, they are (or should be) weighing all the factors RbR suggests (which surely the prosecution would present) in their "reasonable doubt" analysis. But those other factors don't affect the accuracy of the test.
RbR's description also bears some resemblance to an example in Innumeracy, about a case in Los Angeles in 1964. (I'm not saying RbR is guilty of the probabilistic fallacy in that case, just that his description is similar to the set-up for the fallacy, and the story is a fun one, so I'm going to relate it.) Brutally summarized, the argument was: a pair of lawbreakers were seen with several "distinguishing characteristics" (blonde with pony tail with black man with beard in a yellow car). The defendants matched those characteristics. In this case, there was nothing else linking the defendants to the crime, but surely the chances that a couple would match all those characteristics were so small that this must be the couple in question, right?
So it was decided. Until the appeal, in which the defense pointed out that the relevant probability was not whether a couple would have those characteristics, but rather whether, out of the ~2M couples in LA, there was _another_ couple with those characteristics. (The probability that there was one such couple was actually a certainty, since the defendants were living proof.)
It turns out (and no one would suggest that this is intuitive) the likelihood of another such couple (given the prosecution's estimate of the relevant probabilities) is about 8%, which provided reasonable enough doubt for the California supreme court
to overturn the original verdict.
E) If anyone wants to talk more about math, probability, and math education, I'm very keen on it -- you could say it's my job. Email me!
Showing posts with label education. Show all posts
Showing posts with label education. Show all posts
Sunday, September 13, 2009
Monday, June 22, 2009
TNC cannot stop talking about slavery
If you read only 200 things Ta-Nehisi Coates writes this year, this should be one of them. Also, you will only have read about a fifth of the great stuff he writes.
Insightful, as always, but I don't think TNC is quite right here. Or perhaps he is overall, but some nuance is in order.
Specifically, I think there's a historical line that goes like TNC says "We tend to think", namely that slavery was kind of this regrettable cruelty that the world in general was tolerant of, until we got civilized and disavowed it.
But there is at least an acknowledgment, in the historical line I was taught, that America (like always) was exceptional. That line says, what with the explosion of cotton (which Eli Whitney's invention contributed to), the economy of the South became completely dependent on slavery as an economic fuel, kind of like we talk about being dependent on oil today. And that dependence shifted the culture, both toward a vigorous defense of the economic interests in slavery and toward dehumanization of slaves on a scale and to a degree beyond whatever the barbaric practices of yesteryear had been.
But that's just my impression from my history classes. I'm not sure there's a shared "American" conventional point of view on this.
Where I feel TNC has it right is that "we have never grappled with this," `this' being that most of the culture of America is formed from the fractious and unresolved shards that remain from centuries of dehumanizing black people, and the struggles both violent and nonviolent to erode and destroy them.
Again, referring to my own education, the mythology of America is the "melting pot" notion of disparate cultures coming together and peacefully sharing the opportunities of a new land. But this, it seems to me, is a gloss, a "and they all lived happily ever after" resolution of the unresolved issue that defines America. We (which of course is not really "we", since we weren't there, but the imagined "we" of this country) invented a new kind of racism, and have never been able to put that dire genie back into the bottle.
Insightful, as always, but I don't think TNC is quite right here. Or perhaps he is overall, but some nuance is in order.
Specifically, I think there's a historical line that goes like TNC says "We tend to think", namely that slavery was kind of this regrettable cruelty that the world in general was tolerant of, until we got civilized and disavowed it.
But there is at least an acknowledgment, in the historical line I was taught, that America (like always) was exceptional. That line says, what with the explosion of cotton (which Eli Whitney's invention contributed to), the economy of the South became completely dependent on slavery as an economic fuel, kind of like we talk about being dependent on oil today. And that dependence shifted the culture, both toward a vigorous defense of the economic interests in slavery and toward dehumanization of slaves on a scale and to a degree beyond whatever the barbaric practices of yesteryear had been.
But that's just my impression from my history classes. I'm not sure there's a shared "American" conventional point of view on this.
Where I feel TNC has it right is that "we have never grappled with this," `this' being that most of the culture of America is formed from the fractious and unresolved shards that remain from centuries of dehumanizing black people, and the struggles both violent and nonviolent to erode and destroy them.
Again, referring to my own education, the mythology of America is the "melting pot" notion of disparate cultures coming together and peacefully sharing the opportunities of a new land. But this, it seems to me, is a gloss, a "and they all lived happily ever after" resolution of the unresolved issue that defines America. We (which of course is not really "we", since we weren't there, but the imagined "we" of this country) invented a new kind of racism, and have never been able to put that dire genie back into the bottle.
Sunday, November 02, 2008
In a comment to this Matt Yglesias post:
It's like the song says: you've got to be carefully taught.
Hearing that ("You've Got to be Carefully Taught") from the Wee Kids on the Kid Power album is surely before Matt's time. (To be fair, I never saw the show on television, since it was a little before _my_ time too. Still have the album though.)
And that seems like a harsh reminder that progress is won only inch by inch. The same sentiment that was an indictment of racism in 1949, and bore repeating in 1974, is equally appropriate regarding homophobia in 2008.
The more things change, the more ... we must fight for change.
It's like the song says: you've got to be carefully taught.
Hearing that ("You've Got to be Carefully Taught") from the Wee Kids on the Kid Power album is surely before Matt's time. (To be fair, I never saw the show on television, since it was a little before _my_ time too. Still have the album though.)
And that seems like a harsh reminder that progress is won only inch by inch. The same sentiment that was an indictment of racism in 1949, and bore repeating in 1974, is equally appropriate regarding homophobia in 2008.
The more things change, the more ... we must fight for change.
Thursday, August 28, 2008
What Bob does when he should be working
Feel free to follow along with Bob's websurfing -- it's almost like being there!
As you know, I follow Matt Yglesias (who has a new bloghome), and he posted this reference/comment on university education, in which he notes that Brad DeLong, in this "excellent post on the origins of the large lecture course",
So I went and read DeLong's whole post, which is deliciously concise, which doesn't make for really effect procrastinating, so I read on and found my ego vicariously mollified in this comment:
The rest of the websurf into the blogshore is less interesting -- Googling "is iTunes evil?" convinced me that it was, but not so much that I shouldn't install it to hear a (presumably free) podcast of Carl. But I'm not currently logged in with privileges enough to install it, and so then I figured I'd blog my experience for you to share.
The "large lecture" issue doesn't come up at my place o'business, but it does at Rebecca's. But the general concern of "is this the way to teach?" is the Pressing Issue of Our Time. Feel free to put your random thoughts or suggestions in the comments, and maybe you will start a voyage of webby discovery for some other slacker!
As you know, I follow Matt Yglesias (who has a new bloghome), and he posted this reference/comment on university education, in which he notes that Brad DeLong, in this "excellent post on the origins of the large lecture course",
observes that large lectures had a compelling logic in the pre-Gutenberg universe....Modern practice, by contrast, is a bit puzzling.
So I went and read DeLong's whole post, which is deliciously concise, which doesn't make for really effect procrastinating, so I read on and found my ego vicariously mollified in this comment:
Brad, if you are seriously interested in this, you should speak to Carl Wieman....The comment goes on to summarize Carl's take on university ed, and suggests you can hear it direct from him:
You can hear him discuss what he is on about here:This link, it turns out, is on iTunes, which I don't have installed, at least on this machine.
http://phobos.apple.com/WebObjects/MZStore.woa/wa/viewPodcast?id=95882746&s=143441&i=1556149
The rest of the websurf into the blogshore is less interesting -- Googling "is iTunes evil?" convinced me that it was, but not so much that I shouldn't install it to hear a (presumably free) podcast of Carl. But I'm not currently logged in with privileges enough to install it, and so then I figured I'd blog my experience for you to share.
The "large lecture" issue doesn't come up at my place o'business, but it does at Rebecca's. But the general concern of "is this the way to teach?" is the Pressing Issue of Our Time. Feel free to put your random thoughts or suggestions in the comments, and maybe you will start a voyage of webby discovery for some other slacker!
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