I like technological breakthroughs, and the announcement in this article struck me as good news:

Nuclear power plants smaller than a garden shed and able to power 20,000 homes will be on sale within five years, say scientists at Los Alamos, the US government laboratory which developed the first atomic bomb.

The miniature reactors will be factory-sealed, contain no weapons-grade material, have no moving parts and will be nearly impossible to steal because they will be encased in concrete and buried underground.

The US government has licensed the technology to Hyperion, a New Mexico-based company which said last week that it has taken its first firm orders and plans to start mass production within five years. 'Our goal is to generate electricity for 10 cents a watt anywhere in the world,' said John Deal, chief executive of Hyperion. 'They will cost approximately $25m [£13m] each. For a community with 10,000 households, that is a very affordable $250 per home.'

It hardly inspires confidence that the CEO of a company which makes nukes would make such a blatant error in math.

I guess it's possible that the error wasn't his, but was instead that of the writer. You know, a zero got dropped. Does this indicate an "error"? I'm very forgiving of obvious errors, especially typos, but the problem with numerical errors is that the intended meanings aren't always clear the way they are when, say, the word "the" appears as "teh."

The question of whether a number is an error is compounded if we look at the possible biases that might be behind it. In that context, what is an "error"? That a number is simply wrong does not end the inquiry. If I am "just a digit off" when I tell the IRS that I made $25,000 when the number is really $250,000, it is certainly relevant whether I deliberately gave them an incorrect number as opposed to having made a mistake in math or in transcription. Yet the number is "wrong," and in that sense an "error" regardless of my intent. At that point, though, the question of "what is an error" becomes a philosophical one. In the moral and legal sense, if I deliberately gave the IRS an incorrect number, I made no error, for my act was deliberate. Lies are not errors. Yet in math, an error is an error is an error. Suppose that for some pathological reason you take a math exam, and you deliberately gave the wrong answer knowing it was wrong. It is still wrong, and good or bad faith do not affect its inherent wrongness. A wrong answer is no more wrong because it was sincerely believed to be right either. Interestingly, if by your calculations you got the wrong answer, then made a transcription error which made it right, it would still be right. Morality and intent are irrelevant.

Notice that from the facts given in the current example, it is not clear which number is in error. There are three figures:

$25 million cost per plan

10,000 customers

$250 cost per customer.

I can't be sure, but I think the $250 should be $2500. Still affordable. Anyway, the lower the price, the better the news, right?

It's a minor point, but the philosophical aspects of errors fascinate me, and in an amazing coincidence, earlier this morning I read an article dealing with precisely that subject, titled "Math students find success with philosophical route to the right answers":

It seems weird at first: Math teachers who don't care if the answer is correct.

However, it is claimed (by the proponents) that these methods work better to improve test scores:

But Wayne State University professors who teach college math at Detroit Public Schools believe that triumph is not in the answer, it's in the struggle to get there.

Now, their philosophy -- one that test scores suggest works with surprising success -- has found a bigger and more efficient home on WSU's main campus.

The Center for Excellence and Equity in Mathematics is about "creating an environment in which kids are intellectually fearless, where they're not so worried that their answers are incorrect or correct," said math Professor Leonard Boehm.

Boehm is a man in constant motion -- pacing and punching the air in the classroom, shaking students' hands and cracking really bad jokes. Four times a week, he brings college math to fifth-graders at Thurgood Marshall Elementary School, just one of the programs now served by CEEM.

On a recent morning, the wooden floor boards creaked beneath his stride and 26 students were rapt.

The lesson: If 2 raised to the second power is 4, and 2 raised to the third power is 8, what is 2 raised to the zero power?

Hands shot in the air. "My esteemed colleagues," each respondent began their answer. (The answers in Boehm's class always start this way as a sign of respect.)

The students agreed: The answer must be zero, right?

No, wait. 1? Their certainty wavered. 2?

Boehm grinned and stopped. His voice dropped.

"How come a mathematician," he asked, "might say the answer doesn't exist?"

It was Desean Washington-Jones, 11, who eventually raised his hand through the quiet.

The students began gesturing -- like a basketball referee motions for a traveling offense -- to show their support as Desean worked through the answer.

"My esteemed colleagues," he began. "The answer might not exist because there is no factor form."

Hands flew in the air and began to shake -- the sign they agreed.

"Can something come from nothing?" Boehm asked. "If a factor form doesn't exist, can the answer come from something that doesn't exist?"

And so it went -- onto existential quantifiers, the fallacy of induction, and lessons that will lead to the concepts of a limit as well as infinite geometric and harmonic series.

It's the kind of college-prep training not always available to or expected of inner-city kids, said Steve Kahn, WSU professor and director of CEEM.

"The kids in Detroit are getting screwed," said Kahn, who began his career teaching Detroit's high school dropouts.

The numbers cited are impressive:

The CEEM approach seems to be working. Before Math Corps, students answered about 30% of the answers correctly on grade-appropriate tests. By the end of the camp, they were averaging 90%, according to CEEM stats.

"The math is good and simple. ... But the real key is changing a kid's attitude about her or his expectations," said Robert Thomas, dean of WSU's College of Science.

That, and matching students with peer mentors "turns out to be magical," he said.

Assuming that the numbers are correct, I wondered if anything might explain this other than magic, so I went to the Math Corps web site, and saw this:

In reviewing applications, staff seeks indications of a serious desire to succeed in mathematics as well as evidence that the student is willing to work hard to achieve that success.

Of course, if the goal is "math equity" (a complex issue described here) what might be the implications for the test? If the goal is to improve scores to promote equity, that might not translate into real world proficiency. What I'd like to know is how they would do on a standardized test, and whether they end up being able to do things like long division. It's one thing to teach that correct math answers don't matter as an initial stage, but I'd hate to think that they never really matter. Otherwise, how is anyone supposed to make sense out of anything numerical?

But once again, maybe my hangup about accuracy reveals my own educational deficiencies. I was bad at math, and one of the reasons was that there might have been too much emphasis on finding accurate numbers. Here's how I put it before:

...appreciating how many dead bodies there might [as in victims of Hurricane Katrina] be is a highly personal process. To one person, there might be hundreds. To others, there might be thousands, and depending on social skills and psychological considerations, still others might see the answer as millions.

Aren't higher numbers more relevant to what's going on in the world? If the goal of math is to make things relevant, then the numbers have to be higher, because otherwise, people might not care as much.

What this means is that the hangup that bloggers like me have with finding accurate numbers reveals an educational deficiency which is being remedied.

I should be glad. Because it means my being bad at math really wasn't any shortcoming on my part. And my hangup about it only reflects the wrong social attitudes of the times in which I grew up.

This is all changing.

We should be glad.

If there is no accountability, then why are only some are off the hook?