## Wednesday, October 6, 2010

### Stuff I'm Reading

The "19-equal temperament" scale in music, a proposal to divide the octave (doubling of frequency) into 19 near-logarithmic parts. Argued in composer Joel Mandelbaum's PhD thesis that it's the only viable system with a number of divisions on this order of magnitude. Coincidentally matches the Hebrew calendar system and its pattern of leap years.

## Friday, September 24, 2010

## Friday, September 17, 2010

### Morse Code

## Wednesday, September 1, 2010

### Now I Use More Greek Letters

## Tuesday, August 31, 2010

### Permutation Puzzle

## Wednesday, August 25, 2010

### On Corrupt Godless Programmers

This completely short-circuits my usual "angry" filter. Is this a genuinely new idea in the world? Or is this the same as Dungeons & Dragons game religious criticism back in the 80's?

It's all about criminality.

And yes, I will say that coding as an activity does corrupt. I think it's because geeks as a class tend to be godless or agnostic. Sure, you will find the occasional self-professed believing Christian or Muslim or Jew, but by and large, coders do not recognize a Higher Power. They are not People of the Book, because they only recognize their own book, which is code. There are some that realize this manufactured, man-made thing is merely a creation, and not the Creator, and merely a bad imitation of the Creator's works in Nature. But most don't. Most think the coded artifacts are *better*.

This cult of the belief in code-as-law and coders as god particularly infects the virtual world industry, where people get to code not merely some word-processing application or processor of some function on the web, but get to control human beings very visibly, in the round, in 3-D. They love that.

I don't.

I think it's the beginning of their criminality, by which I mean their violations of the law and civilization norms to take, keep, and abuse power.

## Saturday, August 14, 2010

### More Equals Signs

Reported is stuff like: "The two researchers suggest using mathematics manipulatives". I disagree. The problem is not lack of manipulatives. The problem is that nobody ever told students what the fucking equals sign means.

I'm semi-convinced that a greater emphasis needs to be paid on the physical syntax and grammar of writing (and as a result, reading) mathematics by students throughout the education system. But that's me.

## Friday, August 13, 2010

### Zimbabwe

http://en.wikipedia.org/wiki/Hyperinflation_in_Zimbabwe

## Wednesday, August 11, 2010

### Godel's Naturalization

Read it here.

## Tuesday, August 10, 2010

### Monkeynomics

http://scientopia.org/blogs/thisscientificlife/2010/08/10/laurie-santos-how-monkeys-mirror-human-irrationality

The "Take home message of the talk" (as she says at 16:47) is that the choice to take risk differs on whether the situation is perceived as a gain or a loss -- regardless of the risk/reward being exactly the same in each case. When presented with the option of either (a) 2 grapes, or (b) 50/50 chances for either 1 or 3 grapes:

- Monkeys take the safe choice (a) in a gain situation, i.e., start with 1 grape and possibly add some more later,

- Monkeys take the risky choice (b) in a loss situation, i.e., start with 3 grapes and possibly take some away.

That being the same as humans tend to do on analogous tests. My personal interpretation is that this points out how negative numbers are actually a very sophisticated, hard thing to deal with for most people (and other organisms). Most of the time in a natural community you'd be taking actions to gain things -- the "loss" scenario is somewhat artificial and abusive, and we're not set up naturally to deal with that well (i.e., we don't have a natural built-in processor for negatives, and for most brains things just kind of go "kablooey" when forced to deal with them).

## Thursday, July 29, 2010

### Wow, That's Freaking Cheap

Similarly, don't forget about ye olde "Verizon Math Fail" recording from a couple years back. Glad that wasn't me.

## Tuesday, June 8, 2010

### This Is The Dumbest Goddamn Thing You Can Say About Statistics

"A large population size must require a larger sample size."This -- or any iteration thereof -- is the dumbest goddamn thing you can say about statistics. While it's a clear demonstration that someone's missed the whole point of inferential statistics, it's also one of the most common things you'll hear about them. (Often in the form of "That sample is only a small proportion of the population.") Here's some of the varieties of this statement that I've encountered over time:

How do they project statistics like that? I'm trying to imagine what kind of sample size you'd need to represent, well, everything in the universe. [In regard to matter/anti-matter ratio in the universe as researched at Fermilab; comment posted at Slashdot]

Adobe claims that its Flash platform reaches '99% of internet viewers,' but a closer look at those statistics suggests it's not exactly all-encompassing... the number of Flash users is based on a questionable internet survey of just 4,600 people — around 0.0005% of the suggested 956,000,000 total. [News summary at Slashdot]

That poll doesn't convince me of 4e's success or lack thereof. Also, there's only 904 total votes while ENWorld has over 74,000 members, so that's only a small fraction of forum members (addmittedly many of those 74,000 are probably inactive). [In regard to the popularity of the D&D game's 4th Edition; comment posted at ENWorld]You get the idea. To save some writing time here, I'll use n to indicate the sample size and N to indicate the population size. For any statistical inference, if n=50 is an acceptable sample for N=1,000, then it's also acceptable for N=10,000, N=1 billion, or N=infinity. In particular, one thing that never really matters is the ratio of sample to population.

Brief illustration: Let's say that you're using a sample mean to estimate a population mean (much like in a scientific opinion survey, etc.). As long as you have a sample size of at least 30 or so, you automatically know what the shape of all possible sample mean results is: a normal curve, as per the (mathematically proven) Central Limit Theorem. And then you can use that curve (via some integral calculus, or a resulting table or spreadsheet formula) to calculate the probability that your observed sample mean is any given distance from the population mean. Does the size of the population have any bearing on this sampling distribution shape? No. Does the CLT make any reference to the size of the population? No, not whatsoever. You have a moderate-sized sample (30+), you know the shape of all possible sample means, you calculate your probability from that (or some equivalent process), done.

Exception: In calculating sampling distribution probabilities, you'll use something like the fact that its standard deviation is σ/√n. (Here the σ indicates the standard deviation of the whole population.) Now, if the population size happens to be exceptionally small (like, N≤20n), and you're sampling without replacement, then you can improve the estimate a bit by instead using the correction formula √((N-n)/(N-1)) * σ/√n. But why bother? (a) You're almost never in that situation, (b) it rarely makes that much difference, and (c) you're just making extra number-crunching work for yourself. So you're actually better off assuming that the population is really huge or even infinite (as is actually done), thereby saving yourself calculation effort by way of the simpler formula. For any N>20n, the difference is negligible anyway (which is to say: lim

_{N→∞}√((N-n)/(N-1)) * σ/√n = σ/√n). Run some numerical examples (pick any σ you like) and you'll see how little difference it makes.

Even more absurd exception: One requirement that the Central Limit Theorem does have is that the population standard deviation must be nonzero, i.e., σ>0, which does rule out having a population size of just one. But, c'mon, if that were the case then what you're doing isn't really sampling or inferential statistics in the first place, now is it?

In summary: If anything, a larger population size makes the statistics easier, and the math is simplest when you assume an infinite population size in the first place. Other than that, population size has no bearing on the math behind your estimation or surveying procedure.

One final, really simple observation: If an opinion poll is performed at the standard 95% confidence level, then its margin of error can be basically calculated by: E = 1/√n. (Compare to the formula for standard deviation above; the σ disappears due to a particular very convenient substitution and cancellation.) Does the population size N appear anywhere in this formula? Nope -- it's fundamentally irrelevant to the process.

(I've written about this before, but I wanted a version that was a bit more -- ahem -- direct, for posterity's sake.)

## Thursday, June 3, 2010

### Stuff that Shouldn't Work

Another pitfall is creating problems that are singularities, i.e., the correct answer can be produced by some completely incorrect process, one that won't work for any other problem of the same nature. For remedial math students, this is almost a nightmare scenario, since their capacity to correctly generalize from the specific to the abstract is already shaky and confused as it is.

As just one example, here's one of my favorites from the algebra workbook we use at my school (custom edition produced by other teachers in same department):

If 1.05x = 22.05, then x = ?Now, the correct process is to divide both sides by 1.05, and see that x = 22.05/1.05 = 21. But horribly, if a student mistakenly

*subtracts*1.05, then they also get the same answer! Say x = 22.05 - 1.05 = 21. Thus, this exercise allows a student to "submarine" a totally broken process (answers are multiple-choice in the book), giving them apparent confirmation that they're doing the right thing when they're absolutely not. (Note that this particular exercise was changed in a newer edition after I pointed it out.)

Enough prelude. The thing I'm trying to get around to is that last night I saw the "crown jewel" for this kind of problem, as part of a set of practice problems for the ACT Compass Test in Algebra. (You can actually see it here: "Sample Math Test Questions: Numerical Skills/Pre-Algebra and Algebra", Algebra item #14). Something like this:

For x ≠ 3, reduce (xNow, the point of this exercise is to practice factoring (in this case, the top is a "difference of squares") and then cancel like factors on the top & bottom. Write: (x^{2}- 9)/(x-3).

^{2}- 9)/(x-3) = (x+3)(x-3)/(x-3) = x+3.

But last night my students got all weirded out when I was writing that much (there's an additional wrinkle in the Compass problem, but it's not germane to my point) and said they got the right answer with a lot less work. They explained: "Divide x

^{2}on top by x on bottom and get x. Divide -9 on top by -3 on bottom and get +3. There's the answer, x+3."

Now obviously this is a horribly mutilated process (and not uncommon!), thinking that you can divide individual terms in a rational expression piecemeal. (My best explanation, not that it gets fantastic traction, is always "Division distributes across addition, so if you divide by x, you have to divide

*every term*by x." ) But the really crazy unique thing about this problem is that the broken process actually works for

*every possible problem of this format*!

Consider all possible ways of constructing a "difference of squares" on top, and one of its canceling factors on the bottom. Case 1: Say you're reducing (a

^{2}-b

^{2})/(a+b). Correct process: (a

^{2}-b

^{2})/(a+b) = (a+b)(a-b)/(a+b) = a-b. Incorrect process: (a

^{2}-b

^{2})/(a+b) = a

^{2}/a - b

^{2}/b = a-b (same answer). Case 2: Say you're reducing (a

^{2}-b

^{2})/(a-b). Correct process: (a

^{2}-b

^{2})/(a-b) = (a+b)(a-b)/(a-b) = a+b. Incorrect process: (a

^{2}-b

^{2})/(a-b) = a

^{2}/a - b

^{2}/(-b) = a+b (also the same answer).

So not only does the "broken" process work for all permutations of this kind of problem, it even manages to get all the signs correct regardless of how those have been set up. Arrrghh!!!

(Silver lining: At least two of my students had the courage to tell me that's what they'd done, and I had the presence of mind to listen to it last night. I've used this practice test for about 5 years without anyone pointing out how they were doing it like that.)

## Wednesday, May 12, 2010

### Number 91

Consider this: With just three exceptions, all of the composite numbers up to 100 have a factor of either 2, 3, or 5. So, for all of those numbers, you can pretty much immediately see that they're composite, via your simple, standard divisibility-identifying tricks.

The three exceptions, things that have a "7" built into them as their smallest factor, are: 49 (7*7), 77 (7*11), and 91 (7*13). Of course, 49 and 77 are pretty obviously composite (knowing one's squares and what divisibility-by-11 looks like).

So 91 is the only composite number up to 100 that I can't immediately identify. I tend to incorrectly conclude that it's prime if I'm just working at it mentally.

## Thursday, April 29, 2010

### Counterexample to the CLT!?

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.

## Tuesday, April 27, 2010

### Sets as Plastic Baggies

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.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]

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.

## Thursday, April 8, 2010

### ... In New Jersey

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 youHoly 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?addthe 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.

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).

## Thursday, March 25, 2010

### Psych Today Article

http://www.psychologytoday.com/blog/freedom-learn/201003/when-less-is-more-the-case-teaching-less-math-in-schools

## Thursday, March 4, 2010

### Equals Or Not Equals

So on a lot of the homework, inevitably, some fraction of the class

*leaves out all of the equals signs*. And since I go through and just check off each line as being correct or not, this results in every single line being wrong, resulting in

*zero points for the entire assignment*. Even if everything else was correct, even if all the final results are the same as the answer sheet.

So in one sense, kind of brutal. In another sense, this requirement was made abundantly clear in class, including a fully written example from me on the submission page that they're turning it in on. So if you can't follow that simple requirement, one might argue it's an eminently reasonable end result.

But here's another way of looking at it: Each line of math is, effectively, a sentence. (A highly condensed sentence in specialized notation, but the same nonetheless. It can be re-hydrated back into normal English at any time.)

*And the equals sign is the verb "to be"*. It's the most important verb in any language! What if someone were in a writing class and submitted a paper

*without any verbs*? What if they were entirely unable to say "you are", "I am", "he is" anything at all? Would an English teacher totally flip out? You bet they would.

And that's

*exactly*my reaction when I see a paper like that.

Complete sentence: 5 + 3 = 8 ("Five plus three is eight.")

Sentence fragment: 5 + 3 8 ("Five plus three eight.")

Repeat that sentence fragment for 10 or 20 problems per page and see what happens to your eyeballs. Slings and arrows and all that.

## Thursday, February 25, 2010

### Lottery Puzzle

## Thursday, January 28, 2010

### "Community" TV Show

But wait, there's more. I've long held a theory about performance that, "No actor can play more intelligent than they are in real life". Largely this is a matter of diction: I hear actors stumbling over, or putting incorrect emphasis on, pieces of vocabulary that they don't really know or use themselves in daily life. "Community" places really great actors throughout the ensemble; they're delivering lines like "as long as we keep the work/fun ratio the same I want to keep seeing you", and "the deception is making the sex 36% hotter" so fluidly that they slide by almost without me realizing that they were meant to be jokey.

## Tuesday, January 26, 2010

### Elementary Teachers

Beilock, who studies how anxieties and stress can affect people's performance, noted that other research has indicated that elementary education majors at the college level have the highest levels of math anxiety of any college major.

This in a larger article finding:

But by the end of the year, the more anxious teachers were about their own math skills, the more likely their female students — but not the boys — were to agree that "boys are good at math and girls are good at reading." In addition, the girls who answered that way scored lower on math tests than either the classes' boys or the girls who had not developed a belief in the stereotype, the researchers found.

http://news.yahoo.com/s/ap/us_sci_fear_of_figures

**Edit:**The article linked above is gone from Yahoo News. But you can still see that via the Internet Archive's Wayback Machine. And here's a citation and link to the original academic article:

*Proceedings of the National Academy of Sciences*107.5 (2010): 1860-1863. (Link)

## Sunday, January 3, 2010

### Phonics and Bases

When I was very young, we were taught reading by "phonics", i.e., sounding out the letters of unknown words, and then thinking about how those sounds related to words we already knew. Sometime after that, phonics was dropped for a "whole word" approach, but it seems like the pendulum might be swinging back these days.

Why do "phonics" make sense as an instructional strategy?

*Because our system of writing is a technology based on exactly that principle.*The whole

*point*to our alphabet is that it is phonogrammatic, i.e., written symbols represent spoken sounds. This system of writing is

*meant*for there to be an obvious connection between what we write, and what we say. (Of course, this is totally different from logographies such as Chinese whole-word characters and Egyptian heiroglyphics, but that's an entirely different story.) In teaching a child to read, why would you

*not*use the language tool the way it was designed to be used?

Similarly, the traditional way of teaching arithmetic is to memorize a small number of basic facts (addition and multiplication tables), and then learn fundamental written procedures for adding, subtracting, multiplying, etc., large (many-digit) numbers. More recently, we've had to deal with the "Math Wars" is which training in time-tested prodecures has been frowned upon as too authoritarian (or something). Rather, there appears to be extensive time spent in base-10-system conceptual understanding, and then inventive or creative exhortations to make up your own multifarious addition, subtraction, etc., algorithms.

Why do "procedures" make sense as an instructional strategy? Again,

*because our system of written numbers is a technology based on exactly that principle.*The whole

*point*to our place-value system (base-10) is that it is intended to make the specific procedures for adding, subtracting, and multiplying simple, straightforward, and consistent for everyone! Consider the history of written numbers: With ancient systems like, again, Egyptian numerals or Roman numerals, there was

*no way*to get addition or multiplying accomplished by simple writing or mental effort. Without a fixed base system, numbers don't "line up" the way they do for us. To get any arithmetic done (including total sales or tax calculations), you had to go to a licensed counter (think: public notary) with their counting board or abacus tool for use as a mechanical calculator, and trust that the computations they did there were correct. You were entirely at the mercy of this elite, cryptic profession, just to do simple addition.

At some later point, our Hindu-Arabic numeral system was invented and made its way to the West. This system of writing numbers is

*meant*for there to be an obvious connection between the numbers we write, and how to add and multiply them, via a specific written procedure. Kings and princes were astounded at the prospect of people being able to do arithmetic on paper, or in their head, without using an abacus as a calculator. It's like magic! It's not some kind of accident that we use a positional-number system, it was engineered that way

*only*so that we could use a specific adding and multiplying algorithm. In teaching a child to do arithmetic, why would you

*not*use the written number tool the way it was designed to be used?

In summary, it's fascinating that both reading & writing instruction have taken almost exactly parallel paths in the last few decades in America. In each case,

*they have abandoned the rationale for which the writing tool was designed in the first place*. It's like showing someone a power saw for this first time and them asking them, "Can you think of something you might use this for?" To leave out the

*intention*of the tool is to miss the whole

*point*of it. We should play to the strengths of our writing technology, and not frustrate ourselves fighting against it.

### Math Wars

http://en.wikipedia.org/wiki/Math_wars

Basically, there's been a dispute over whether to emphasize "procedural" (algorithmic, memorized step-by-step processes) or "conceptual" (creative, inventive, big ideas) skills in the earliest grades. In the last 2 decades or so the "conceptual" camp has basically won the debate in teacher education schools, claiming to have research backing up the approach. Recently there have been calls for a more middle-ground approach.

Interesting articles in this month's

*American Educator*magazine. One by cognitive psychologist Daniel T. Willingham: http://www.aft.org/pubs-reports/american_educator/issues/winter09_10/willingham.pdf

In cultivating greater conceptual knowledge, don't sacrifice procedural or factual knowledge.Procedural or factual knowledge without conceptual knowledge is shallow and is unlikely to transfer to new contexts, but conceptual knowledge without procedural or factual knowledge is ineffectual. Tie conceptual knowledge to procedures that students are learning so that the "how" has a meaningful "why" associated with it; one will reinforce the other. Increased conceptual knowledge may help the average American student move from bare competence with facts and procedures to the automaticity needed to be a good problem solver. But if we reduce work on facts and procedures, the result is likely to be disastrous.

And another article by professor E.D. Hirsh: http://www.aft.org/pubs-reports/american_educator/issues/winter09_10/hirsch.pdf

The victory of the progressive, anti-curriculum movement has chiefly occurred in the crucial early grades, and the further down one goes in the grades, the more intense the resistance to academic subject matter with the greatest wrath reserved for introducing academic knowledge in preschool. It does not seem to occur to the anti-curriculum advocates that the four-year-old children of rich, highly educated parents are gaining academic knowledge at home, while such knowledge is being unfairly withheld at school (albeit with noble intentions) from the children of the poor. For those who truly want equality, a common, content-rich core curriculum is the only option. It is the only way for our disadvantaged children to catch up to their more advantaged peers.