My apologies to those out there who have already seen this confusing bit o' math trickery:
The resolution to this little conundrum? If you want to know, you'll have to click here for a related discussion. In a way, the above algebra is actually a proof of why division-by-zero is undefined!
Did you know there is more than one proof of how 0.999999... (repeating) equals exactly 1?
No calculus involved, just simple algebra. See it for yourself here.
In a previous post, I derided Ray Kurzweil for showing a straight line on a log-log graph and calling that an "exponential trend." As I said in that post:
...the math savvy should recognize that a straight line on a log-log graph is not indicative of exponential growth but instead illustrates a power law relationship. So does that chart contradict the exponential developments that are the cornerstone of his arguments?
Well, as it turns out, the chart does not contradict the exponential trend. You might ask, how can you get an exponential trend to look like a straight line on a log-log graph? The answer is very subtle, actually. Let's take a simple model of exponential growth:
Where n is something like number of inventions, and t is time (the other two parameters, n0 and k, are there just for the sake of generalization). The key to making this a straight line in log-log space is to plot the number of inventions vs. the time until next invention. With this subtle change, the plot becomes linear on a log-log plot. The time until next invention, τ(t), is fairly simple to obtain:
When you plot τ(t) vs. n(t), you get a straight line. So I was wrong, and Kurzweil was right. I still maintain that the way Kurzweil plots this is very deceptive. The reason that he didn't plot it on a normal semilog plot (like all the others he normally shows), is that the scatter would look huge. By compressing both axes into log scale, it makes it look as if the correlation is better than it really is; as far as I'm concerned, this is a rather big no-no for serious academics. I think many peer-reviewers would recoil upon seeing data plotted in such a way.
Just my two cents.
I was just reading the news this morning when I stumbled across this headline and the corresponding article summary. Anything seem odd to you?
In case you didn't notice it, the BBC makes two very different statements: (1) that the total number of HIV infections may have peaked, and (2) that the world's HIV infection rate may have peaked. This confusion in their headline is analogous to saying a car stopped moving when in reality it just stopped accelerating. Of course nobody's hypothesizing that HIV infections might actually start to drop off (although it's tempting to think that based on the headline). Once again we see the importance of having basic math skills, no matter what Richard Cohen says about its importance in journalism.
Derivatives, people. Derivatives...
UPDATE (2005-05-31): Today's New York Times gets it perfectly right:
We Yanks may not get cricket, but at least we understand math.
I heard about Richard Cohen's recent algebra-bashing article in the Washington Post entitled "What is the Value of Algebra?", then I finally read it.
At the Science Blogs, comments like "lame," "stupid," and "outrageous" were pretty standard. It's a difficult situation to comment on, but made only more difficult by the language Cohen uses to disparage math education.
In the article, Cohen takes up the story of Gabriela Ocampo (a name eerily close to the Dallas news reporter, Gloria Campos), who dropped out of the 12th grade after failing algebra six times. Without giving any relevant details of her particular situation or home life, he sides against Gabriela's teachers, saying:
Most of math can now be done by a computer or a calculator. On the other hand, no computer can write a column or even a thank-you note — or reason even a little bit. [...] If, say, the school asked you for another year of English or, God forbid, history, so that you actually had to know something about your world, I would be on its side. But algebra? Please.
Gabriela, sooner or later someone's going to tell you that algebra teaches reasoning. This is a lie propagated by, among others, algebra teachers. Writing is the highest form of reasoning.
And, of course, the coup de grâce:
In L.A., more kids drop out of school on account of algebra than any other subject. I can hardly blame them.
Cohen's utter ignorance makes my blood boil. Not only does he fulfill every (bad) stereotype of C. P. Snow's other culture, he is siding with ignorance. Regardless of how he tries to frame it, he really is defending ignorance by defending walking out on education. This makes me wonder how Cohen would react to an engineer telling young students that they don't have learn to communicate effectively in order to get a job.
One is tempted to think that Cohen's article is an attempt at over-the-top satire, like suggesting for the starving to eat their own children. But even after re-reading it twice, I don't think it is...
Promoting ignorance is not the right way, it never is. How can Richard Cohen call himself a liberal?