Counterexample to the CLT!?

The introductory statistics text that I teach out of presents the Central Limit Theorem this way:

The Central Limit Theorem (CLT): For a relatively large sample size, the variable x' is approximately normally distributed, regardless of the distribution of the variable under consideration. The approximation becomes better with increasing sample size. [Weiss, "Introductory Statistics" 7E, p. 341]

In other words, any distribution turns into a normal curve when you're sampling (with large sample sizes). I also know off the top of my head that the formal CLT is talking about a distribution that's been standardized (converted z = (x-μ)/(σ/√(n))), and how its limit as a function is the standard-normal curve (centered at 0, standard deviation 1).

So one day I'm walking around sort of half-dozy and I'm thinking, "Wait a minute! What about a constant function? If you had a distribution that was fixed with one element (say, 100% chance that x=5), any conceivable sample mean would just be the constant x'=5, and there's no way a graph of that looks like a normal curve, right?"

Meditating...

Well, the thing that I didn't immediately have in my head, and is also entirely left out of the Weiss text -- There is one single fine-print requirement to the CLT, and it's that the standard deviation of your variable must be nonzero (and also non-infinite), i.e., 0<σ<∞. Which is sort of obvious from the fact that you need that to standardize with z = (x-μ)/(σ/√(n)), it being used to divide with in the formula and all. And of course a constant function has zero deviation, so it's indeed outside the scope of the theorem.

Guess I can't get too mad at the Weiss text for this... chances of it being useful for an introductory student to spend time parsing that is about nil. (Obviously, it hasn't come up in 5+ years of teaching the class.) Still, it might be nice to put it in a little footnote at the bottom of the page so I don't go daydreaming about possible counterexamples on my commute.

Sets as Plastic Baggies

Many times I've seen the use of set-braces (roster notation) compared to an "envelope". Here's one example (speaking of empty sets and the null-set symbol):

To help clarify this concept, think of a set as an envelope. If the set is empty, then the envelope is empty. On the other hand, if the set is not empty -- that is, it contains at least one element -- then there are items in the envelope. One such item can be another envelope. Using this analogy, the symbol {Ø} specifies an empty envelope contained within another envelope. [Setek and Gallo, "Fundamentals of Mathematics" 10th Ed., p. 74]

Now, what I think is the jarring discordance in this analogy: You can immediately see the contents inside a { } symbol, but not so with envelopes (being opaque and all). That's probably why the whole metaphor always feels foul in my mouth, and might be part of the reason I get a poor reaction from students when I try to use it in class.

Let's try a better metaphor: A clear plastic baggie (with a zip, perhaps?). These you can, like the set { } symbol, instantly see into. If you put one plastic baggie inside another, then you can immediately see it sitting inside there... just like the frequently-misused {Ø} notation.

So let's use a metaphor that shares in the transparency and clarity of our precise math notation.

... In New Jersey

So I got a few inquiries about the prior post on how a Psychology Today article could make me yell out loud. Yes, I was sitting in my apartment alone and hollered out, "Oh NOO!!!" at this part:

In an article published in 2005, Patricia Clark Kenschaft, a professor of mathematics at Montclair State University, described her experiences of going into elementary schools and talking with teachers about math. In one visit to a K-6 elementary school in New Jersey she discovered that not a single teacher, out of the fifty that she met with, knew how to find the area of a rectangle.[2] They taught multiplication, but none of them knew that multiplication is used to find the area of a rectangle. Their most common guess was that you add the length and the width to get the area. Their excuse for not knowing was that they did not need to teach about areas of rectangles; that came later in the curriculum. But the fact that they couldn't figure out that multiplication is used to find the area was evidence to Kenschaft that they didn't really know what multiplication is or what it is for. She also found that although the teachers knew and taught the algorithm for multiplying one two-digit number by another, none of them could explain why that algorithm works.

Holy god, that is insane. I have a hard time imagining anyone being unable to find the area of a rectangle, never mind school staff actually teaching arithmetic, to say nothing of going 0-for-50 in a survey on the subject. I mean... unbelievable! Maybe I should take a poll of students in one of my own classes. Is it possible that people had just forgotten what the word "area" meant?

Just to complete the progression in the article:

The school that Kenschaft visited happened to be in a very poor district, with mostly African American kids, so at first she figured that the worst teachers must have been assigned to that school, and she theorized that this was why African Americans do even more poorly than white Americans on math tests. But then she went into some schools in wealthy districts, with mostly white kids, and found that the mathematics knowledge of teachers there was equally pathetic. She concluded that nobody could be learning much math in school and, "It appears that the higher scores of the affluent districts are not due to superior teaching but to the supplementary informal ‘home schooling' of children."

On the larger thesis of the article, that current math instruction in K-6 is doing more damage than good, and could and has been dropped successfully at least once... you know what? I can potentially believe that. It's possible. If the quality of math instruction is truly that atrocious, I wouldn't want children subjected to it -- of course the only consistent result would be crippling lifelong math anxiety (per Dijkstra's, "as potential programmers they are mentally mutilated beyond hope of regeneration," and all that).

But far more important than my own best-guesses would be: This would take a lot more research before going forward with a plan of that nature. And no way in hell would we either get (1) the research approved, or (2) the program implemented here in the USA (what with the READING AND MATH UBER ALLES!! approach to education that comes down from the political wing these days).