Governor Pat Quinn of Illinois just signed a law which abolishes the death penalty in that state. This legislation is to some extent symbolic, in that Illinois has had a moratorium on the death penalty for almost a decade. With this legislation, Illinois joins New Mexico and New Jersey as states that have recently abolished the death penalty.
I have discussed here before my attitude towards the death penalty, especially in regards to specific cases such as the ongoing case of Hank Skinner . I'm not intrinsically against the death penalty: societies have the right in general to execute those who have violated the social contract in particularly heinous fashions if that will assist the public good. However, the death penalty as practiced in the United States is capricious and disorderly. Prosecutors push for death sentences when it is politically convenient, and their are huge racial disparities in who is executed.
It is also clear that innocent people have been wrongly convicted, and that in at least some cases, innocent people have been executed such as Cameron Todd Willinngham. Evidentiary standards that allow junk science and superficially persuasive eyewitness testimony are leaving blood on all our hands. For a long time, I have found striking a certain section in the Biblical book of my namesake, Joshua. the Israelites are called thieves when one man steals at Jericho. How much worse are we as a society that we as a democracy repeatedly elect and reelect officials who kill innocent people in our name?
In the particular case of the state of Illinois, I have additional personal reason to be interested in this legislation. My uncle Seymour Simon was a justice on the Illinois Supreme Court. He was a staunch opponent of the death penalty and argued forcefully and unceasingly that the death penalty as implemented in Illinois was unconsciousable and unconstitutionally capricious. He died in 2006, and so did not live to see this legislation. I suspect I know how he would have responded if he had seen this: Illinois is down. Only thirty-five more states to go.
Wednesday, March 9, 2011
Thursday, February 24, 2011
Space Shuttle Discovery launch
I'm currently in Florida with friends watching the launch of Discovery. This will be the second to last shuttle launch (there's a small chance that this will be the third to last launch if they get enough funding for a final launch). I and Tyrol5 wrote an article for Wikinews about the launch which includes a picture taken by me of the launch. There are additional photos on my Facebook page.
Monday, February 14, 2011
On rotations of spheres
Consider a sphere living in some number of dimensions. The sphere we are used to is the 2-sphere, denoted by S2. Mathematicians call it this because even though it lives in 3-dimensions, it is itself a 2-dimensional object (in the sense that small sections of it look like the 2-dimensional plane). One can similarly talk about the n-sphere, which lives in n+1 dimensions. Thus for example, the 1-sphere, S1, can be thought of all all points on a plane that are of distance one from the origin.
When one has a geometric object one of the most obvious things to do is to ask what rigid movements of the object will take it to itself. Thus, for example, for a sphere, a rotation about some axis through the sphere's center rigidly moves the sphere to itself.
Rotations seem simple, but they can be surprisingly tricky. For the 1-sphere if one does a rotation and then another rotation one is left with a rotation. This is an important and helpful property. It says essentially that rotations form what mathematicians call a subgroup of the group of all rigid motions. It turns out that this is still true for the 2-sphere, S2 even when one uses different axises for the two rotation Now, you might find surprising the fact that for S3, the sphere of three dimensions living in four dimensions, this breaks down. The composition of rotations is not necessarily a rotation. In some sense the not obvious fact is not why this breaks down for three dimensions, but why it still holds true in two dimensions.
The above is well known to mathematicians or to many people who have played around a bit with geometric objects. But, I recently learned a related fact that I found startling. Call a rotation "periodic" if we eventually get every point back to where we started. So for example, if one repeat a 90 degree rotation (π/2 for those using radians) four times we will have every point back where we started. Now, it turns out that for rotations of S2 even though the composition of rotations is a rotation, the composition of periodic rotations is not necessarily periodic. Once one knows that this is true it is easy construct examples. Consider two rotations around perpendicular axises each of 30 degrees (π/6 in radians). It isn't difficult to show that their composition although still a rotation is not periodic. This is a good example of how even basic geometry can surprise us even when we think we understand it.
When one has a geometric object one of the most obvious things to do is to ask what rigid movements of the object will take it to itself. Thus, for example, for a sphere, a rotation about some axis through the sphere's center rigidly moves the sphere to itself.
Rotations seem simple, but they can be surprisingly tricky. For the 1-sphere if one does a rotation and then another rotation one is left with a rotation. This is an important and helpful property. It says essentially that rotations form what mathematicians call a subgroup of the group of all rigid motions. It turns out that this is still true for the 2-sphere, S2 even when one uses different axises for the two rotation Now, you might find surprising the fact that for S3, the sphere of three dimensions living in four dimensions, this breaks down. The composition of rotations is not necessarily a rotation. In some sense the not obvious fact is not why this breaks down for three dimensions, but why it still holds true in two dimensions.
The above is well known to mathematicians or to many people who have played around a bit with geometric objects. But, I recently learned a related fact that I found startling. Call a rotation "periodic" if we eventually get every point back to where we started. So for example, if one repeat a 90 degree rotation (π/2 for those using radians) four times we will have every point back where we started. Now, it turns out that for rotations of S2 even though the composition of rotations is a rotation, the composition of periodic rotations is not necessarily periodic. Once one knows that this is true it is easy construct examples. Consider two rotations around perpendicular axises each of 30 degrees (π/6 in radians). It isn't difficult to show that their composition although still a rotation is not periodic. This is a good example of how even basic geometry can surprise us even when we think we understand it.
Wednesday, February 2, 2011
Obama's State of the Union Address: A Response to Nathaniel
President Obama delivered his State of the Union address last week. My younger brother has a piece in the Yale Daily News which extensively criticizes the speech. Nathaniel complains that the speech started with an inspirational message but "soon fell into a quagmire of policy." Nathaniel criticizes the speech for focusing too much on policy minutia and argues that, while referencing Kennedy, Obama fails to understand that inspiration requires large goals, not policy details. Overall, Nathaniel is correct that this was a disappointing speech. However, it was not disappointing because it went into detail: it was disappointing because the details were unimpressive.
While Nathaniel is correct that the speech was not very inspirational, he misses a broader point: We need policy expertise, and a President who is willing to present a State of the Union that discusses real policy issues is a good thing. After 8 years of George W Bush's flag-wrapping jingoism, after 8 years of Bill Clinton's contentless addresses, the American people should be happy that a President is willing to discuss serious policy issues. And if the American people don't like that, that's a problem with the American people, not a problem with their President.
Nathaniel has two other criticisms of the speech: that Obama was unwilling to take a stand on anything remotely controversial, and that Obama called for largescale spending that the recent elections show is not desired by the American people. These criticisms seem contradictory. One cannot in one paragraph complain that Obama was a coward and then in the next paragraph complain that Obama is too willing to engage in spending that many people don't want.
Some parts of Nathaniel's column do have merit. Nathaniel correctly identifies that Obama's comments about school reform were close to toothless. And Nathaniel correctly points out that Obama did not emphasize the degree to which the proposed initiative will place heavy burdens on the American people. And it seems disingenuous for Obama to claim to be willing to fund the " Apollo Projects of our time" while calling for a freeze in domestic spending.
While I disagree with much of Nathaniel's criticism, I am far from happy with Obama's speech. It certainly had its good points, but it had many failings. Obama failed badly in his discussion of energy policy. He gets points for mentioning nuclear power, something he has in the past downplayed. However, when discussing biofuels, he made no mention of the fact that the most prominent attempt at biofuel, corn based ethanol, is inefficient, environmentally damaging, and raises food prices which hurts the poor here as well as people in the developing world.
Similarly, Obama's emphasis on electric cars was similarly unpersuasive. The President did not discuss that electric cars’ power has to come from the general grid. While electric cars do overall pollute less and use less energy, this stems from economies of scale more than anything else. And the impact is not high. While Obama did discuss the energy of the future, he proposed no funding for research into genuinely new energy sources, such as fusion power. (In the particular case of fusion power, the US is putting resources into ITER, the international tokamak reactor, but the US is putting no money into other forms of fusion such as stellarators.)
Obama's discussion of school systems used metrics which are less than ideal. In particular, the fraction of a population which is going to college is an awful metric for success of students. Students who go to college are often unprepared and often come out not much better prepared than they went in. Moreover, many jobs, even high-tech jobs, don't require college degrees. We have serious problems with people going to college on loans they are unable to repay.
Obama was correct to talk about policy proposals in his address. The problem is not the discussion but the content of those policies.
While Nathaniel is correct that the speech was not very inspirational, he misses a broader point: We need policy expertise, and a President who is willing to present a State of the Union that discusses real policy issues is a good thing. After 8 years of George W Bush's flag-wrapping jingoism, after 8 years of Bill Clinton's contentless addresses, the American people should be happy that a President is willing to discuss serious policy issues. And if the American people don't like that, that's a problem with the American people, not a problem with their President.
Nathaniel has two other criticisms of the speech: that Obama was unwilling to take a stand on anything remotely controversial, and that Obama called for largescale spending that the recent elections show is not desired by the American people. These criticisms seem contradictory. One cannot in one paragraph complain that Obama was a coward and then in the next paragraph complain that Obama is too willing to engage in spending that many people don't want.
Some parts of Nathaniel's column do have merit. Nathaniel correctly identifies that Obama's comments about school reform were close to toothless. And Nathaniel correctly points out that Obama did not emphasize the degree to which the proposed initiative will place heavy burdens on the American people. And it seems disingenuous for Obama to claim to be willing to fund the " Apollo Projects of our time" while calling for a freeze in domestic spending.
While I disagree with much of Nathaniel's criticism, I am far from happy with Obama's speech. It certainly had its good points, but it had many failings. Obama failed badly in his discussion of energy policy. He gets points for mentioning nuclear power, something he has in the past downplayed. However, when discussing biofuels, he made no mention of the fact that the most prominent attempt at biofuel, corn based ethanol, is inefficient, environmentally damaging, and raises food prices which hurts the poor here as well as people in the developing world.
Similarly, Obama's emphasis on electric cars was similarly unpersuasive. The President did not discuss that electric cars’ power has to come from the general grid. While electric cars do overall pollute less and use less energy, this stems from economies of scale more than anything else. And the impact is not high. While Obama did discuss the energy of the future, he proposed no funding for research into genuinely new energy sources, such as fusion power. (In the particular case of fusion power, the US is putting resources into ITER, the international tokamak reactor, but the US is putting no money into other forms of fusion such as stellarators.)
Obama's discussion of school systems used metrics which are less than ideal. In particular, the fraction of a population which is going to college is an awful metric for success of students. Students who go to college are often unprepared and often come out not much better prepared than they went in. Moreover, many jobs, even high-tech jobs, don't require college degrees. We have serious problems with people going to college on loans they are unable to repay.
Obama was correct to talk about policy proposals in his address. The problem is not the discussion but the content of those policies.
Thursday, January 6, 2011
Review of Jason Rosenhouse's "The Monty Hall Problem"
I've had the recent pleasure of reading Jason Rosenhouse's "The Monty Hall Problem." Rosenhouse's book is a comprehensive investigation into the eponymous Monty Hall problem, variations of the problem, and the larger implications of the problem.
The original Monty Hall problem named after a game played on an old television game show "Let's Make a Deal" with host Monty Hall. Rosenhouse describes the problem as:
(Presumably you are attempting to maximize your chance of winning one's chance of getting a car). Most people conclude that there's no benefit from switching. The general logic against switching is that after the elimination of a door there are two doors remaining, so each should now have a 1/2 chance of containing the door.
This logic is incorrect. One door has a 2/3rds chance of getting the car if one's general strategy is switching. Many people find this claim extremely counterintuitive. To see quickly the correctness of this claim, note that if one chooses a strategy of to always switching, then one will switch to the correct car-containing door exactly when your original door was not the car door. This will occur 2/3rd of the time.
Many people have great difficulty accepting the correct solution to the Monty Hall problem. This includes not just laypeople, but also professional mathematicians, including most famously Paul Erdos who initially did not accept the answer. The problem, and variants thereof, not only raise interesting questions of probability but also give insight into how humans think about probability.
Rosenhouse's book is very well done. He looks not just at the math, but also the history of the problem, and philosophical and psychological implications of the problem. For example, he discusses studies which show that cross-culturally the vast majority of people when given the problem will not switch. I was unaware until I read this book how much cross-disciplinary work there had been surrounding the Monty Hall problem. Not all of this work has been that impressive, and Rosenhouse correctly points out where much of the philosophical argumentation over the problem simply breaks downs. Along the way, Rosenhouse explains such important concepts as Bayes' Theorem (where he uses the simple discrete case), the different approaches to what probabilities mean (classical, frequentist, and Bayesian) and their philosophical implications. The book could easily be used for supplementary reading for an undergraduate course in probability or reading for an interested highschool student.
By far the most interesting parts of the book were the chapters focusing on the psychological aspects of the problem. Systematic investigation of the common failure of people to correctly analyze the Monty Hall problem has lead to much insight about how humans reason about probability. This analysis strongly suggests that humans use a variety of heuristics which generally work well for many circumstances humans run into but break down in extreme cases. In a short blog post I can’t do justice to the clever, sophisticated experimental set-ups used to test the nature and extent of these heuristics, so I'll simply recommend that people read the book.
For my own part, I'd like to use this as an opportunity to propose two continuous versions of the Monty Hall problem that to my knowledge have not been previously discussed. Consider a circle of circumference 1. A point is randomly picked as the target point on circle (and not revealed to you). You then pick a random interval of length 1/3rd on the circle. Monty knows where the target point is. If you picked an interval that contains the target point, Monty picks a random 1/3rd interval that doesn't overlap your interval and reveals that interval as not containing the target point. If your interval does not contain the target point, Monty instead picks uniformly a 1/3rd interval that doesn't include the target point and doesn't overlap with your interval. At the end of this process, you have with probability one, three possible intervals that might contain the target point, your original interval, or the intervals on either side of Monty's revealed interval. You are given the option to switch to one of these new intervals. Should you switch and if so to which interval?
I'm pretty sure that the answer in this modified form is also to switch, in this case switching to the larger of the two new intervals.
However, the situation becomes a bit trickier if we modify it a bit. Consider the following situation that is identical to the above, but instead of Monty cutting out an interval of length 1/3rd, he picks k intervals of each length 1/(3k) (thus the initial case above is k=1). Monty picks where to place these intervals by each picking one of the valid intervals uniformly and then going on to the other, then revealing the locations of all his intervals at the end. The remaining choices for an interval for you to pick are your original interval or any of the smaller intervals created in between Monty's choices. You get an option to stay or to switch to one of these intervals. It seems clear that even for k=2, sometiimes you should switch and sometimes you should not switch, depending on the locations of Monty's intervals. However, it isn't clear to me when to stay and when to switch. Thoughts are welcome.
The original Monty Hall problem named after a game played on an old television game show "Let's Make a Deal" with host Monty Hall. Rosenhouse describes the problem as:
You are shown three identical doors. Behind one of them is a car. The other two conceal goats. You are asked to choose, but not open one of the doors. After doing so, Monty, who knows where the car is, opens one of the two remaining doors. He always opens a door he knows to be incorrect, and randomly chooses which door to open when he has a more than one option (which happens on those occasions where your initial choice conceals the car). After opening an incorrect door, Monty gives you the option of either switching to the other unopened door or sticking with your original choice. You then receive whatever is behind the door you choose. What should you do?
(Presumably you are attempting to maximize your chance of winning one's chance of getting a car). Most people conclude that there's no benefit from switching. The general logic against switching is that after the elimination of a door there are two doors remaining, so each should now have a 1/2 chance of containing the door.
This logic is incorrect. One door has a 2/3rds chance of getting the car if one's general strategy is switching. Many people find this claim extremely counterintuitive. To see quickly the correctness of this claim, note that if one chooses a strategy of to always switching, then one will switch to the correct car-containing door exactly when your original door was not the car door. This will occur 2/3rd of the time.
Many people have great difficulty accepting the correct solution to the Monty Hall problem. This includes not just laypeople, but also professional mathematicians, including most famously Paul Erdos who initially did not accept the answer. The problem, and variants thereof, not only raise interesting questions of probability but also give insight into how humans think about probability.
Rosenhouse's book is very well done. He looks not just at the math, but also the history of the problem, and philosophical and psychological implications of the problem. For example, he discusses studies which show that cross-culturally the vast majority of people when given the problem will not switch. I was unaware until I read this book how much cross-disciplinary work there had been surrounding the Monty Hall problem. Not all of this work has been that impressive, and Rosenhouse correctly points out where much of the philosophical argumentation over the problem simply breaks downs. Along the way, Rosenhouse explains such important concepts as Bayes' Theorem (where he uses the simple discrete case), the different approaches to what probabilities mean (classical, frequentist, and Bayesian) and their philosophical implications. The book could easily be used for supplementary reading for an undergraduate course in probability or reading for an interested highschool student.
By far the most interesting parts of the book were the chapters focusing on the psychological aspects of the problem. Systematic investigation of the common failure of people to correctly analyze the Monty Hall problem has lead to much insight about how humans reason about probability. This analysis strongly suggests that humans use a variety of heuristics which generally work well for many circumstances humans run into but break down in extreme cases. In a short blog post I can’t do justice to the clever, sophisticated experimental set-ups used to test the nature and extent of these heuristics, so I'll simply recommend that people read the book.
For my own part, I'd like to use this as an opportunity to propose two continuous versions of the Monty Hall problem that to my knowledge have not been previously discussed. Consider a circle of circumference 1. A point is randomly picked as the target point on circle (and not revealed to you). You then pick a random interval of length 1/3rd on the circle. Monty knows where the target point is. If you picked an interval that contains the target point, Monty picks a random 1/3rd interval that doesn't overlap your interval and reveals that interval as not containing the target point. If your interval does not contain the target point, Monty instead picks uniformly a 1/3rd interval that doesn't include the target point and doesn't overlap with your interval. At the end of this process, you have with probability one, three possible intervals that might contain the target point, your original interval, or the intervals on either side of Monty's revealed interval. You are given the option to switch to one of these new intervals. Should you switch and if so to which interval?
I'm pretty sure that the answer in this modified form is also to switch, in this case switching to the larger of the two new intervals.
However, the situation becomes a bit trickier if we modify it a bit. Consider the following situation that is identical to the above, but instead of Monty cutting out an interval of length 1/3rd, he picks k intervals of each length 1/(3k) (thus the initial case above is k=1). Monty picks where to place these intervals by each picking one of the valid intervals uniformly and then going on to the other, then revealing the locations of all his intervals at the end. The remaining choices for an interval for you to pick are your original interval or any of the smaller intervals created in between Monty's choices. You get an option to stay or to switch to one of these intervals. It seems clear that even for k=2, sometiimes you should switch and sometimes you should not switch, depending on the locations of Monty's intervals. However, it isn't clear to me when to stay and when to switch. Thoughts are welcome.
Wednesday, December 8, 2010
New Haven Schools (again), The Death Penalty, and Gratuitous Promotion of Family Members
Two quick notes:
Nathaniel has another piece criticizing the New Haven Promise program. This piece says much of what he has already said but includes a few more direct proposals about what else should be done. Unfortunately, this piece is on National Review Online. While that's an impressive accomplishment, that does make me feel conflicted about linking to it. In general, while NRO does have some worthwhile material, a disturbingly large fraction of it has over the last few years come to resemble a nutshell of what is wrong with the modern conservative movement in the US. For example, considering that NRO is the same place where writer Mike Potemra complained that Star Trek promoted "peace, tolerance, due process, progress" which are much too liberal values. Aside from this issue, as I've discussed before, I think that the problems in the public schools, including New Haven, are more complicated than Nathaniel portrays them.
There is a piece by another member of my family that I can link to with fewer reservations. My father has a deeply personal piece up at the Oxford University Press blog discussing why he has grown to oppose the death penalty. I suspect that his changing views are similar to the general decline in support for the death penalty over the last 20 years. The general support for the death penalty has dropped over time although about two thirds of the US still supports the death penalty (I've seen claims that the US did not have strong support for the death penalty in the 1960s and that the level of support grew for some time before beginning its current downwards trend, but I've never seen data backing this claim up.) There's also evidence that more religious people tend to be more likely to support the death penalty. If that is a causal rather than correlative link, the current drop in support for the death penalty may be due to the general drop in popularity of organized religion over the last few years.
Nathaniel has another piece criticizing the New Haven Promise program. This piece says much of what he has already said but includes a few more direct proposals about what else should be done. Unfortunately, this piece is on National Review Online. While that's an impressive accomplishment, that does make me feel conflicted about linking to it. In general, while NRO does have some worthwhile material, a disturbingly large fraction of it has over the last few years come to resemble a nutshell of what is wrong with the modern conservative movement in the US. For example, considering that NRO is the same place where writer Mike Potemra complained that Star Trek promoted "peace, tolerance, due process, progress" which are much too liberal values. Aside from this issue, as I've discussed before, I think that the problems in the public schools, including New Haven, are more complicated than Nathaniel portrays them.
There is a piece by another member of my family that I can link to with fewer reservations. My father has a deeply personal piece up at the Oxford University Press blog discussing why he has grown to oppose the death penalty. I suspect that his changing views are similar to the general decline in support for the death penalty over the last 20 years. The general support for the death penalty has dropped over time although about two thirds of the US still supports the death penalty (I've seen claims that the US did not have strong support for the death penalty in the 1960s and that the level of support grew for some time before beginning its current downwards trend, but I've never seen data backing this claim up.) There's also evidence that more religious people tend to be more likely to support the death penalty. If that is a causal rather than correlative link, the current drop in support for the death penalty may be due to the general drop in popularity of organized religion over the last few years.
Sunday, November 14, 2010
Public Schools, Unions, New Haven Promise, and GPFM
Last week, my alma mater Yale University announced that the university would work together with New Haven to fund "New Haven Promise," a program which would provide funding to New Haven public school students who attend colleges in Connecticut. The program promises scholarships for New Haven public school students with only a few weak restrictions. For example, students with less than 90% attendance rates in highschool are not eligible.
There are a number of possible criticisms of this program. The most serious criticism to me seems to be the simple one that this program is not Yale's job. Alumni donate money to Yale with certain expectations. They might also donate money to other causes. But there is a basic expectation that money that goes to Yale will be used for Yale purposes such as going to scholarships for poor students at Yale, not to students at random other schools in Connecticut.
There are additional problems with this program. My little brother wrote an op-ed in the Yale Daily News arguing that this program would in fact cover up the real issues in the New Haven public school systems which need to be addressed. He argues that the teacher unions and the lazy and incompetent teaching which they allow are much more of a root cause of the problems. I'm not convinced of his claims. I'm especially unconvinced by his line that "Instead of staying after school to tutor or help run an extracurricular, unionized teachers typically leave as soon as the final bell rings" which seems to underestimate the great difficulty that even hard-working teachers need to put up with daily.
However, I do think that more broadly speaking there's a clear problem with unions in our public schools which prevent the removal of all but the most egregiously bad teachers. For example, consider the case of eighth grade science teacher John Freshwater in Mount Vernon, Ohio. Freshwater taught science so badly that other teachers in later years had to specifically reteach Freshwater's students. Freshwater told students that Catholics were not real Christians. Freshwater burned crosses into students' arms using a tesla coil. Despite all these issues, it has taken more than 2 years to have Freshwater removed. Thus, while I don't have enough detailed experience to personally evaluate whether the unions are a problem in New Haven (although my limited anecdotal evidence suggests that they are a problem), it does fit the general pattern of what is going wrong with American public schools.
Nathaniel's piece generated a variety of responses such this one by a New Haven public school teacher, this one by a New Haven alderman, and this one by another Yale student. Nathaniel has responded to the last piece here. All of these pieces are worth reading.
There are a number of possible criticisms of this program. The most serious criticism to me seems to be the simple one that this program is not Yale's job. Alumni donate money to Yale with certain expectations. They might also donate money to other causes. But there is a basic expectation that money that goes to Yale will be used for Yale purposes such as going to scholarships for poor students at Yale, not to students at random other schools in Connecticut.
There are additional problems with this program. My little brother wrote an op-ed in the Yale Daily News arguing that this program would in fact cover up the real issues in the New Haven public school systems which need to be addressed. He argues that the teacher unions and the lazy and incompetent teaching which they allow are much more of a root cause of the problems. I'm not convinced of his claims. I'm especially unconvinced by his line that "Instead of staying after school to tutor or help run an extracurricular, unionized teachers typically leave as soon as the final bell rings" which seems to underestimate the great difficulty that even hard-working teachers need to put up with daily.
However, I do think that more broadly speaking there's a clear problem with unions in our public schools which prevent the removal of all but the most egregiously bad teachers. For example, consider the case of eighth grade science teacher John Freshwater in Mount Vernon, Ohio. Freshwater taught science so badly that other teachers in later years had to specifically reteach Freshwater's students. Freshwater told students that Catholics were not real Christians. Freshwater burned crosses into students' arms using a tesla coil. Despite all these issues, it has taken more than 2 years to have Freshwater removed. Thus, while I don't have enough detailed experience to personally evaluate whether the unions are a problem in New Haven (although my limited anecdotal evidence suggests that they are a problem), it does fit the general pattern of what is going wrong with American public schools.
Nathaniel's piece generated a variety of responses such this one by a New Haven public school teacher, this one by a New Haven alderman, and this one by another Yale student. Nathaniel has responded to the last piece here. All of these pieces are worth reading.
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