AngryMath

Against Factoring Trees

2012-01-29T08:00:00.000-05:00

Nowadays, I'm anti-factoring trees. It's funny, because they're not usually part of the classes I teach, but they've come up a few times recently -- including twice just yesterday (as I write this), when they were included in a new book I received, and then within an hour a student came asking about them (because they were part of a YouTube lecture on reducing radicals she'd been trying to watch).

By factoring trees, I mean the method for producing the unique prime factorization of a number that looks like this:
By which we can conclude (re-sorting the leaf nodes) that 48 = 2^4 * 3. Of course, this is pretty customary, and it's how pretty much everyone I know (including myself) learned how to do it.

But now my primary complaint against them is that they're a nonstandard method of writing mathematical relationships, and most of all, they're a lost opportunity to practice writing equality statements. With all the problems that students have using, writing, and understanding the equality sign, why not use this as an opening to reinforce their meaning -- especially so in a context just like this, where we do not intend to simplify (evaluate) on the right hand side? Why not instead write in a more standard format like this:

(Or whatever your preference is for use of parentheses or exact number of steps.) It highlights all these issues with the meaning of equality signs that we struggle with later on students' behalf, and it avoids using a special one-off writing technique for the singular task of factoring a number. It's likely easier to read for some students (who may have trouble identifying where the leaves of the tree are). It even saves on lines of paper, and is easier to type out in an email or website if you have to do that. This is actually what I do in class when it comes up now. The more mental connections we can make to the "correct" way of writing math, the better.

PEMDAS Poetry, Pt. 1

2012-01-20T00:15:00.002-05:00

Like a drunk uncle
Stumbling towards the bathroom door
Mistakes will happen

More anti-PEMDAS proselytizing here.

Comments on Decimal Places

2012-01-15T08:00:00.000-05:00

Previously observed -- Operations on powers with the same base effectively shift one place down in the order of operations. Examples:

Exponentiation will multiply powers; e.g., (x^3)^4 = x^12
Multiplication will add powers; e.g., x^2 * x^3 = x^5
Addition does a no-op on powers: e.g., 3x^2 + 5x^2 = 8x^2

Note that the same rule generally applies regarding number of places after a decimal point. Examples (working in the other direction):

Addition does not alter number of places; e.g., 1.2 + 3.4 = 4.6 (1 place)
Multiplication adds the number of places; e.g., 1.2 * 3.46 = 4.152 (3 places)
Exponentiation multiplies the number of places; e.g., 1.234^2 = 1.577756 (6 places)

And the reason, of course, is that the decimal places themselves represent exponents to the common base 10. Example:

1.2 + 3.4
= (1*10^1 + 2*10^-1) + (3*10^1 + 4*10^-1)
= (1*10^1 + 3*10^1) + (2*10^-1 + 4*10^-1)
= 4*10^1 + 6*10^-1
= 4.6

Since the number of decimal places is dictated by the lowest power, if the addition operation 2*10^-1 + 4*10^-1 doesn't change the power (as above), then neither will it change the number of decimal places. And so on and so forth.

Calculator Equals

2012-01-08T08:00:00.001-05:00

On the subject of students not understanding equals signs -- probably not the first time someone pointed this out, but -- How much of this is caused by usage of the equals sign button on a calculator?

It's really a bit malformed, if you think about it. Mathematically what's really happening when you hit that button is a request to "simplify" the numerical expression that you've typed in so far. So perhaps it would be better if the button were labelled "simplify" or "evaluate" -- or maybe a "total" button like on cash registers, or some abbreviation along those lines.

Possibly the malformed understanding of the equals symbol (thinking that a simplified number always goes on the right side) is due to the hundreds and thousands of times that students have used a calculator "=" button by the time the issue matters in algebra?

Related posts:

Concrete P-Value Demonstration

2012-01-01T08:00:00.000-05:00

I find that students in my statistics class are almost totally bewildered by the logic of hypothesis testing and P-values (for hypotheses based on a population mean), no matter how carefully I try to explain the concepts. Here's an idea for a super-short and simple, concrete demonstration of hypothesis testing. Tell me if you think this would be worth the class time:

Start with a hand of four cards: {A, 2, 3, 4}
I'll turn my back and secretly do one of two things:
H₀: Leave the Ace in, or
H_A: Take the Ace out
Now shuffle the hand and deal out 3 cards.

Question: Say I get a draw of {2, 3, 4}. What's the chance of this happening if I did not take out the Ace (H₀)? Note that all possible draws would be {{A,2,3}, {A,2,4}, {A,3,4}, {2,3,4}} so the probability of seeing that would be P = f/N = 1/4 = 0.25.

Conclusion: If I draw {2,3,4} then we have some evidence that I did change the deck (H_A) -- because it's unlikely to see that result if I didn't (P = 0.25).

Now -- You can actually demonstrate this and ask the class if they think I left the Ace in or took it out each time. I'd recommend 3 run-throughs: leave it, leave it, then take it out. (In the latter case, also ask: Is it possible that I left the Ace in?) In reality, you should probably hold the cards against the otherwise full box, so it isn't obvious if your hand becomes empty in the take-it-out case. (And otherwise practice the prestidigitation in advance so your handwork doesn't give it away.)

Open Question: Should I actually reveal to the class which one I did each time (for confirmation), or leave that as a mystery (modeling real-world usage)?

The Gdmn Particle

2011-12-25T08:00:00.001-05:00

Another fabulous anecdote from the RJLipton Blog, regarding the Higgs Boson (what's popularly called "The God Particle"):

Higgs himself believes neither the particle nor the mechanism should carry his sole name, and was happy that he, Brout, Englert, and the three authors of another 1964 paper (Gerald Guranik, Carl Hagen, and Tom Kibble) were all awarded the 2010 J.J. Sakurai Prize for this work. He may have gotten his wish, as the popular name “The God Particle” has stuck to the boson. This is the title of a 1993 book by Nobel prize-winning physicist Leon Lederman and science writer Dick Teresi.

According to Higgs, Lederman had wanted to title the book The G*d*mm Particle to emphasize how elusive the boson was. His publisher declined to have a swear word in the title, but thought it fine to use just “God.” However, they could have settled on the Orthodox Jewish practice of writing “G-d” to avoid situations where the fully-written name might be erased or discarded. The title The G-d Particle could then be read with Lederman’s original meaning or not. Higgs is said to join many scientists regretting the “God Particle” name, more from concern over hype than irreverence.

I love this story so much. First, it finally makes sense of that stupid name in a way that eluded me until now. Secondly, it again shows that the "real" existential experience of scientific problem-solving is more generally one of a desperate, teeth-grinding, curse-filled battle (and not so much a dainty and refined observation of museum-like beauty).

Divine grace is a marketing pitch you use to sell something to the public. It's not something seen in the real world, or actual live math work, very much.

On that note, happy holidays from AngryMath! :-)

Dyson Quote

2011-12-18T08:00:00.000-05:00

An excellent AngryMath-approved quote from Freeman Dyson:

If science ceases to be a rebellion against authority, then it does not deserve the talents of our brightest children.

The Peanut Butter Protocol

2011-11-14T08:00:00.002-05:00

Here's something that pops up in math/computer science that I honestly just HATE so much (I got sufficiently riled up while describing it to my girlfriend tonight that I thought it would make a perfect blog post). On the question of "How do you introduce programming concepts to students for the very first time (possibly children)?", a very common answer is "Ask them to give the steps for making a peanut-butter and jelly sandwich!" (or something very similar). For example, whenever this comes up on Slashdot, the responses are predominantly along the lines of "love this... lovely... hilarious" (link). But I'm completely contrarian about it.

Of course, the point is basically a "gotcha" exercise: the students say "scoop out the peanut butter" and you go "what!? look, now I'm batting the jar-top with my hand, because you didn't tell me to pick up the knife, and you didn't tell me to screw off the jar-top," etc., etc. etc. If I was a student, and this my first encounter with computer programming, then it would instantaneously sour me on the whole subject, maybe permanently: the task is inherently ambiguous, impossible, unfair, and a trick to apparently set up the respondents for ridicule and embarrassment.

The primary problem (in my opinion) is that's very much not how mathematics or computer programming work. What we must do in practice is to start with an agreed-upon set of atomic operations, which we may call "definitions" or "axioms" or a "function library", depending on the context. Of course, the power of your elementary pieces is variable, depending on the abstraction level at which you're operating. But the real work of programming or proof-building is in how we connect these well-known (and well-defined) basic building blocks in a way that constructs something new, useful, and interesting.

So the "peanut butter sandwich" task is thoroughly and painfully unfair without presenting the allowed operations up front: Am I supposed to say "pick up the knife" or "wrap your fingers around the knife, apply opposing force with thumb, lift forearm" or "bend index finger 5 degrees, now 10 degrees, now 15 degrees..." (it's sort of irrelevant, because without well-defined operations, the presenter can always pick some lower-granularity abstraction and create a "gotcha!" moment). The demonstration does manage to get across the idea that you will be "working with small operations", and also that "unexpected bugs will happen" -- but in my mind, neither of those are essential or even very important. The essence of any creative work is in taking well-known basic tools and building something greater from them than previously existed, and that's something that I think almost anyone can understand and justifiably take satisfaction from.

(P.S. a counter-offer: Rudimentary programming like LOGO. Write on the board 3 allowed operations: (1) turn left, (2) turn right, and (3) step forward. Now direct me how to get from one corner of the room, around some desks, and out the door -- possibly listing the whole instruction set in advance of testing it. Something like that.)

Arguing Infinite Decimals

2011-11-07T01:40:00.004-05:00

Recently the RJLipton blog had two interesting and contentious posts about people who dispute Cantor's diagonal argument (that real numbers have different cardinality than natural numbers), which I'm pretty sure generated more comments than anything else to date on the blog. Apparently this is one of the more popular topics for math-cranks to extensively argue that they've proven the other way -- read for yourself here and here.

I wish that I had the opportunity to address issues like this in the classes I teach, but unfortunately at the moment I don't have any such opportunity. It would be nice to have a venue to refine the argument with a fresh audience every so often, and to work to ferret out the criticisms that arise. If we do so, with a disputatious subject like this (namely: the first few times a student deals with infinite sets and their counterintuitive by-products), then I think it's extra-important that we carefully lay out initial definitions at the start, break down the argument into very atomic numbered steps (so that we can refine discussion and disputes as they come up later), and also give explicit justifications for each step.

Here's another issue which I feel has the same flavor to it: the fact that 0.999... = 1 (or more generally, that any terminating decimal has two different, equivalent representations: the normal one, and a second one that ends with an endless sequence of "9"'s). Here's a suggestion on the careful way that I'd want to do it (again -- not having had this battle-plan encounter the enemy yet):

Definition of 0.999...
(a) The number has infinitely repeating digits.
(b) After every "9" digit, there is another "9".
(c) There is no end to the "9"'s.

Proof that 0.999... = 1 (by algebra)
(1) Let x = 0.999...
(2) Then 10x = 9.999... (multiply each side by 10)
(3) So 9x = 9 (subtract step 1 from step 2; note decimals cancel)
(4) Which means x = 1 (divide each side by 9)
(5) Therefore 0.999... = 1 (substitute from step 1)

And then when the arguments arise you can at least ask your interlocutor to focus on one single step or definition in which they think there's a logical gap.

Statistics vs. the Lottery

2011-08-10T08:23:00.003-04:00

So apparently there's an article in Harper's (can't see the original; link below is commentary at Forbes) on the following subject -- Joan R. Ginther has been "outed" as a statistics professor with a PhD from Stanford, who possibly deduced the winning-ticket lottery distribution schedule in Texas, and has hit multi-million dollar jackpots 4 times in the last decade. Notes:

(1) While I don't see any assertion of any way in which this would be illegal, the overall tone is clearly one of how-dare-she-think-she-can-get-away-with-this. “When something this unlikely happens in a casino, you arrest ‘em first and ask questions later,” says a professor at the Institute for the Study of Gambling & Commercial Gaming at the University of Nevada, Reno.

(2) "The residents of Bishop, Texas seem to believe God was behind it all."

http://blogs.forbes.com/kiriblakeley/2011/07/21/meet-the-luckiest-woman-in-the-world/

Less Time to Learn

2011-07-22T01:46:00.008-04:00

Hypothesis: The less time students have to learn, the higher their testing scores are.

This has been a suspicion of mine for a while now. For example, I find that my accelerated summer/winter modules (6-week courses) generally outperform my normal fall/spring modules (12-week courses) in the subject material, testing procedures, etc. I'm guessing that the major factors involved are (a) a greater focus and more connections with the given subject material, (b) fewer competing courses being taken at the same time, vying for mental attention, and (c) simply less time and opportunity to forget stuff from class to class, which I feel is a real issue for many of my students. (Countering factor might be: Maybe more dedicated students register for summer/winter courses?)

So this summer I had an excellent accidental experiment in this regard. I'm teaching two statistics classes in parallel on Mon/Wed and Tue/Thu nights. There was a weird burp in the schedule (specifically, the Mon Jul-4 holiday) that caused one class to be ahead of the other by one evening's lecture. So heading into the last test (partly on hypothesis tests and P-values), the Mon/Wed class was first introduced to the subject just 2 weekdays (48 hours) in advance of the test, whereas the Tue/Thu class had a whole week (7 days) to see P-values and study for the test (including, obviously, a whole weekend).

So I was rather concerned that the Mon/Wed class was being unfairly put upon, what with such a short window in which to study, and on Wednesday they did seem to struggle. But then to my surprise it turned out that the Tue/Thu class found what was basically the same test even more challenging, and got a significantly lower average score on the same assessment.

On Tau

2011-07-07T14:40:00.010-04:00

So recently there were some popular news articles with titles like, "Mathematicians Want to Say Goodbye to Pi" -- first I've heard of it, and of course initially it sounded ridiculous (I guess that's the point of news-article title-writing, eh?) The gist of it is that in theory, when dealing with circles, it would easier to exchange the value pi = circumference/diameter for tau = circumference/radius, i.e., tau = 2*pi.

And actually, that very quickly hit me as something that would be very nice to have. It would make a lot of trigonometry and calculus easier. The number of radians in a circle would simply be tau (instead of 2*pi). Perhaps most important for me, circles are inherently defined by their radius (all points a given distance from the center), not by their diameter.

Now my first attempt at an objection was the formula for a circle's area, which would get ever-so slightly more complicated, switching from A = pi*r^2 to A = tau*r^2/2. But that's a small thing, and in fact it reminds you of the fundamental integral(r)=r^2/2 which is used to derive it in calculus (instead of a disappearing denominator trick, canceled by the constant 2*pi).

The other thing that just occurred to me -- and motivated this post -- is what it does to Euler's identity, e^(i*pi) = -1 (or however you want to move the terms around). Now, I may be an angry crank, but if I think deeply about this celebrated identity (it was voted "most beautiful formula" in the Mathematical Intelligencer, 1990; a post which I have taped on the wall over my computer), it's not terribly interesting; granted that the imaginary part of the exponential function is a rotation in the complex plane, and coincidentally pi happens to be half a circle, i.e., landing on the point (-1, 0). If we used tau more commonly, then the triviality would be more apparent: e^(i*tau) = 0, and no one would get as worked up about it anymore. Or maybe people would think it's even more "beautiful" then, hell, I don't know. :-)

Am I going to try to switch the thousands-year legacy of using pi to tau? Not me, man, I've got enough to do without quixotic crusades. But yeah, if I could pick different historical legacies the options for (1) switch pi to tau, and (2) switch electrical current signs (link), would be near the top of the list.

What do you think?

Edit: Of course, e^(i*tau) = 1 (not 0). [Knocks self on head.] Maybe that actually is more beautiful.

Math in the Internet Age

2011-06-05T15:59:00.002-04:00

From Mike Jones, commenting on the proposed proof to the Collatz Conjecture:

The paper runs to 32 pages and we will have to wait for it to be checked for errors. Such is mathematics in the Internet age - no longer are proofs brought down from the mountain top in their perfection but they are thrown to the crowd to survive being torn apart.

Lindley's Paradox

2011-03-28T05:06:00.003-04:00

The Wikipedia description of Lindley's Paradox asserts an example of opposite hypothesis-testing results between the Frequentist approach and the Bayesian approach.

The example is one of testing a certain town for the ratio of boy-to-girl births. The thing that violently strikes me here is the choice of the Bayesian prior: P(theta = 0.5) = 0.5, i.e., the advance assumption that it's 50% likely for the ratio to be equal to 0.5 (the other 50% chance spread uniformly between all points from 0 to 1).

I mean: What? Why would I conceivably assume that? If I broadly picture real numbers as being continuous, then my instinct would be to assume that it's almost impossible for any given number to be exactly the parameter value, i.e., I'd assume P(theta = 0.5) = 0. Even if I didn't reason that way, I otherwise have copious evidence that human births aren't really 50/50, there's very clearly more boys born than girls -- so if anything I'd choose that as the most likely prior value.

Is that really how Bayesians are supposed to choose their prior? (It seems atrocious!) Or is this just a fantastically mangled example at Wikipedia?

The Amazing Lottery

2011-03-28T05:02:00.004-04:00

Stats observation of the day -- After every lottery you can say, "That number had only 1 chance in 175 million of coming up!". But, there's a 100% chance you can say that, every time it's run. It's only interesting or significant if you can predict the result in advance. Otherwise you have a fallacy called "data dredging": http://en.wikipedia.org/wiki/Data_dredging

See also:

Other probabilistic fallacies.

Frequentism and LLN

2011-02-28T00:14:00.007-05:00

I would be seriously keen to find this out: What difference is there between the frequentist interpretation of probability and simply a restatement of the Law of Large Numbers? Because I kind of can't see any. And why is the LLN never brought into any such discussion of probability interpretations?

Hacking Secret-Ballot Elections

2011-02-22T16:24:00.001-05:00

One of the good reasons I see argued for secret-ballot elections is, "It prevents employers, union bosses, etc., from demanding a certain vote and then verifying it afterward."

Although in the modern era, you could step into a booth with a ballot, fill it out, take a private camera-phone shot of it with your license in view, and then be required to present that photo to your employer, et. al. There's nothing to prevent that at my polling place, for example. Just a thought.

Thoughts and Cavalry

2011-02-10T01:43:00.002-05:00

Alfred North Whitehead, An Introduction to Mathematics (1911):

By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that … a large proportion of the population of Western Europe could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility … Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation. [...] By the aid of symbolism, we can make transitions in reasoning almost mechanically, by the eye, which otherwise would call into play the higher faculties of the brain. [...] It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilisation advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.

Conjunction Junction

2011-02-06T19:30:00.001-05:00

So the other night while I was recovering from class, I wound up on YouTube watching the "Conjunction Junction" video, from the Schoolhouse Rock series that was broadcast after Saturday-morning shows on ABC all through the 1970's and 1980's. I was kind of surprised by how irritated it made me in one small detail.

I've grown accustomed to teaching the standard logical operators in all of my classes -- whether they be fundamentals of math, introduction to computers, statistics and probability, etc. So when the song starts and says, "I got three favorite cars/ That get most of my job done", well, of course I expected to see "And/ Or/ Not" -- but then what actually appears (as you can see in the top picture) is "And/ But/ Or". So, I was surprised at how jarred I was by that.

As the song explains each connecting word, it first says, "And: That's an additive, like 'this and that'". Okay, makes sense. Then the next bit is: "But: That's sort of the opposite, 'Not this but that'". (See picture below.)

Wait a minute, that's not right! The truth is, the word "But" has the exact same logical meaning as "And" (both clauses are true); all it does is put an interpretive spin on the latter clause, as if to say "this second part may be somewhat surprising to you". And in fact, in order to make the argument that "that's sort of the opposite", they had to go and use the missing logical operator than actually does make things "opposite", namely "Not".

So stick that in your smokestack, Schoolhouse Rock! (But -- I still have the danged thing stuck in my head...)

Watch video here. Or: Read lyrics here.

New Blog Tagline

2011-02-04T01:31:00.000-05:00

You may notice up above I've got a new tag-line ("blog description") under the title of my blog above. This is something one of my better students said at the end of class tonight (making up a lost day of statistics from snow days here in NYC).

"It's like you took a bat and clubbed us with math"; which I think is entirely delightful to think about.

Naming Large Numbers

2011-02-02T13:04:00.001-05:00

A very small observation I'll throw out: Sometimes I want to name a large number that pops up on my calculator in scientific notation (like while lecturing in class, for example). A fast way to do that is to take the exponent and:

Divide by 3
Subtract 1
Say that in Latin (with "-illion").

Now, obviously this requires you to do a few elementary operations in your head and to also know Latin (or at least how to count therein). And: I'm doing this in the "short scale", of course.

Example #1: 1.01238 x 10^18.
Do 18/3-1 = 6-1 = 5. So this is about "one quintillion".

Example #2: 2.13129 x 10^48
Do 48/3-1 = 16-1 = 15. So this is about "two quindecillion".

Example #3: 6.38733 x 10^95
Do 95/3-1 = 31-1 = 30. Since we had a remainder of 2 at the division step, I'll say this is about "six hundred trigintillion".

Grants & Remediation

2011-01-27T02:34:00.000-05:00

A FAQ on New York State TAP (Tuition Assistance Program) grants says this:

Can I get TAP for remedial courses? Remedial courses may be counted towards either full-time or part time enrollment for TAP purposes. However, to qualify for TAP, you must always be registered for a certain number of degree credit courses. [http://www.cuny.edu/admissions/financial-aid/grants-scholarships/nys-grants.html]

Now, why would you want to incentivize taking degree-credit courses when someone hasn't yet completed necessary remedial courses (i.e., prerequisites thereof)? In fact -- require full-time registration for credit courses prior to paying for remedial courses? Especially so when we know half or more of such students won't graduate from the program? (Link.)

That seems quite backwards. Complete the basics first (remediation), then qualify for funding for credit courses afterward.

Stuff I'm Reading

2010-10-06T04:19:00.000-04:00

"Preferred numbers" in engineering and product design; standardized near-logarithm increments to make both interchanging parts and mental arithmetic as easy as possible. (I've liked 1-2-5 for some time, myself.)

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.

One-Day Calculus Lecture

2010-09-24T14:22:00.000-04:00

Dan's brief one-day calculus lecture (1-page PDF): link.

Morse Code

2010-09-17T16:47:00.000-04:00

Visited the USS Intrepid aircraft carrier museum yesterday (here in New York City). Among the numerous exhibits in the hangar deck is this one on Morse Code. The scrolling digital blue lines are supposed to be translations of each other. What's rather glaringly wrong with this?

Now I Use More Greek Letters

2010-09-01T08:40:00.000-04:00

Here's one of the full-page ads currently running on the Virgin Mobile USA home page. Thanks, you jackasses.

Permutation Puzzle

2010-08-31T17:34:00.000-04:00

Among the things I'm not good at are combinatorics and whatnot -- Let's say you have a fixed set of 18 numbers all in the range from 1 to 6 (so obviously a bunch of duplicates), and you need to put 3 of the numbers in each of 6 ordered bins. How many ways can the totals in all the bins come out?

On Corrupt Godless Programmers

2010-08-25T03:22:00.000-04:00

There's a kerfuffle around Second Life at the moment and some shady antics of 3rd-party clients that are officially allowed to connect to the game. That I wouldn't care about, except that it motivated one of the fiercer critics to come up with this novel argument: computer programming (especially 3D) is inherently a godless and corrupt activity.

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.

More Equals Signs

2010-08-14T11:00:00.000-04:00

Previously I wrote about students not using equals signs properly. So apparently some guys at Texas A&M are getting papers published on this subject, and identifying it as a key way to distinguish between high-functioning and low-functioning math students and national education systems.

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.

Zimbabwe

2010-08-13T09:51:00.001-04:00

Did you know -- Annualized inflation in Zimbabwe, late 2008, was estimated to be: 89.7 sextillion percent?

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

Godel's Naturalization

2010-08-11T02:36:00.000-04:00

Back in college I heard this ridiculously awesome story of what happened when the mathematicians Morgenstern, Godel, and Einstein went for Godel's citizenship test. Just ran into a recollection of the event written by Morgenstern (type starts p. 2):

Read it here.

Monkeynomics

2010-08-10T18:52:00.000-04:00

Laurie Santos giving a TEDTalk on the results of an economic experiment with a "monkey market":

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).

Wow, That's Freaking Cheap

2010-07-29T23:21:00.000-04:00

Seen on the ride into school.

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

This Is The Dumbest Goddamn Thing You Can Say About Statistics

2010-06-08T01:37:00.000-04:00

"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.)

Stuff that Shouldn't Work

2010-06-03T16:44:00.000-04:00

Making practice or test exercises is harder than you might first think before becoming a teacher. If you ever make a problem up on the fly while lecturing, it's highly probable that you'll create something with hideous fractions, irrational or imaginary numbers, extraneous solutions, etc., that you didn't want, which winds up sidetracking you from the point you were trying to make.

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 (x² - 9)/(x-3).

Now, 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: (x² - 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 x² 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 (a²-b²)/(a+b). Correct process: (a²-b²)/(a+b) = (a+b)(a-b)/(a+b) = a-b. Incorrect process: (a²-b²)/(a+b) = a²/a - b²/b = a-b (same answer). Case 2: Say you're reducing (a²-b²)/(a-b). Correct process: (a²-b²)/(a-b) = (a+b)(a-b)/(a-b) = a+b. Incorrect process: (a²-b²)/(a-b) = a²/a - b²/(-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.)

Number 91

2010-05-12T15:21:00.000-04:00

What's wrong with the 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.

Counterexample to the CLT!?

2010-04-29T13:20:00.000-04:00

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

2010-04-27T14:20:00.000-04:00

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, 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

2010-04-08T20:04:00.000-04:00

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).

Psych Today Article

2010-03-25T17:19:00.001-04:00

Any article that has me yelling out loud while I read it must be a good one. Highly recommended for several reasons:

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

Still Amazed

2010-03-04T02:59:00.001-05:00

Oh, and I got to do this again today. Still amazed.

Equals Or Not Equals

2010-03-04T00:48:00.000-05:00

In my remedial algebra class, I assign these stupendously short homeworks, but require precisely written work -- professional format, one full operation per line, justification in words at each step. (I don't have much evidence that it helps the students, but at one point last year I had an emotional meltdown trying to read students' normal math writing, so there you go -- it's quite literally defending my own sanity.)

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.

Lottery Puzzle

2010-02-25T17:22:00.000-05:00

Let's say there's a lottery run once every month that costs $100 to enter. Chance of winning is 1 in 20,000. The prize is 5 million dollars. Should you play this lottery?

"Community" TV Show

2010-01-28T17:59:00.000-05:00

The NBC comedy "Community" gets way more things right than they have to. For example: The sexpot statistics professor.

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.

Elementary Teachers

2010-01-26T01:28:00.000-05:00

Did you know?

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

Phonics and Bases

2010-01-03T13:55:00.000-05:00

Speaking of the "Math Wars", here's an observation I made about teaching the most basic elementary-school subjects.

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

2010-01-03T01:28:00.000-05:00

Recently I've had some discussions about the wacky stuff being taught in elementary math classes these days. Not something I deal with directly in the college classes I teach, but turns out there's a whole history to the current situation actually referred to as the "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.

Security Questions

2009-12-24T23:54:00.000-05:00

Consider this. At online finance and banking-type sites, "Your security is important to us." In addition to standard login-id and password, for quite some time they've been fond of using these additional "security questions that only you will know".

Back in the day, it was always one thing in particular: "Mother's maiden name?" Obviously, only you will know that, because it's not important for anything. Well... except that NOW it's important because it got used everywhere as a security question. So every bank I dealt with knows it because they required it for me to do business with them.

So now that's been basically dropped, and a whole slew of other security questions have popped up. "Mother's date of birth?" "Childhood pet's name?" "Where did you go on your honeymoon?" (These are are all actual examples.) Obviously good security questions because no one would want to know any of this trivia.

HEY SECURITY DUMBASS -- AS SOON AS YOU ASK THIS QUESTION IT BECOMES OF INTEREST TO AN ATTACKER, AND THEREFORE A SECURITY VULNERABILITY.

What really pisses me off is that over time, these financial and business sites are going to know every scrap of personal information about my life if this goes on. All my relatives' and friends' birthdays. Nicknames and pets, favorite books/ authors/ places I dream of vacationing, etc., etc., etc. Every time one becomes somewhat widespread, they have to switch to something even more esoteric and private.

Nowadays I'm running into multiple sites (that I've used in the past) that are refusing to allow me access unless I give them some new tidbits of "security question" information. The nice girls at my local bank see my distress and helpfully suggest "Just make something up!" Which has the disadvantages of (a) now I'm not going to remember it and need to write it down, and (b) the fine print of the terms-of-service demand honest and factual information, and while I'm sure the tellers at the bank don't mind, I'm equally sure that the corporate entity will be happy to crucify me over a transgression like that if we ever get into a dispute.

Fuck that.

Disk Icons

2009-12-11T01:24:00.000-05:00

Something else that occurred to me teaching computers recently: Applications still use a picture of a floppy disk to indicate the "save" operation. (See the top of the MS Office Ribbon, in the last post.) This in an era when some of my college students have, apparently, never actually seen a floppy disk. I realized working with some of my students at the end of the semester that this icon doesn't have any intrinsic meaning to them. What to replace it with?

MS Office Ribbons

2009-11-14T14:26:00.000-05:00

So for the first time the computer literacy class I teach has been forced to switch over to MS Office 2007 products, and hence this past week I was finally forced to use the MS Office Ribbon interface. I tried to stay away from it, but now here it is finally. I really don't like it.

Here's the thing I really don't like about (aside from just the radical change from anything that's come before): It's semi-impossible to describe to someone else (say, a student) where they should be clicking.

In some sense, this is basically the primary disadvantage of the GUI having been doubled in intensity. With a command-line interface, it's easy to write out your instructions and transmit them in writing or verbally to another person. The GUI makes that piece of business a lot harder (now the geographic location that you're clicking on makes a big difference, and you wind up clumsily describing the pictures and icons that you're trying to activate).

So with the MS Office Ribbon, this is even more exacerbated. At least with the traditional menu bar, there was a linear order to each step of a process. Click on one menu item, another linear list drops down, find one item in that list, proceed to the next, etc. For example, to center the contents of a cell in Excel, I could provide a handout that says, "Click on: Format > Cells > Alignment > Horizontal > pick 'Center'". But now, I have to say something like "On the Home Tab, find the Alignment section, and kind of near the middle of that section there's a button that kind of has its lines centered, click on that". Very, very clumsy... and more so for lots of other examples that we can probably think of.

Programming Project Idea

2009-10-08T00:54:00.001-04:00

Here's a random class programming project idea. Everyone submits their name and a newly-made up password (not one that they use for anything else). Then, everyone writes a program to guess passwords, lets them run for an hour against the list, and sees how many matches they can make. Suggest a few strategies like a "brute force attack", a "dictionary attack" (providing a dictionary text file), guessing that some people use no numbers or caps (or all numbers), etc. Afterwards, analyze both successful attacks and the more secure passwords.

This would be more advanced than anything I've done in my classes, even though some of the programs could be relatively short. Interesting both for basic programming skill and insights on password security. Maybe seed the list with some instructor-made weak passwords as a baseline target.

Game Theory Lectures

2009-07-29T11:00:00.000-04:00

I've been watching some Game Theory video lectures out of Yale University ( http://oyc.yale.edu/economics/game-theory ). It's a subject area I've been intrigued by for a while, so I was happy to see their availability online. Professor Ben Polak is an energetic and compelling presenter.

However, I keep being reminded that this is a course being given in Yale's Economics department. And I've long held a few very key critiques about the foundations of standard economic theory, that I feel make the entire enterprise miserably inaccurate. What I didn't expect is for these Game Theory lectures to feature a high-intensity spotlight directly on those shortcomings, in practically every single session.

Critique #1 is that economics deals only with money, and wipes out our capacity to deal with other values. Critique #2, probably more important, is that economics fatally depends on a “rational actor” assumption for all involved, which is simply not true. Let's consider them in order:

Critique #1: Economic theory is all about money, and the widespread use of the theory destroys our other values like family, community, craftsmanship, healthy living, emotional satisfaction, and good samaritanhood. As one wise man said, “They don't take these things down at the bank,” and therefore, they get obliterated when economic theory is put in play.

Now, Professor Polak makes a good, painstaking show in Session #1 of trying to fend off this criticism. “You need to know what you want,” he says, and runs an extended example wherein, if one player really was interested in the well-being of his partner in a game, well, that could be accounted for by assessing the value of those feelings, and adding/subtracting to the payoff-matrix appropriately, and then running the same Game Theory analysis on the new matrix, finally arriving in a different result. (Of course, along the way he also snidely refers to this caring player as an “indignant angel"). See? Game Theory can handle all kinds of different values, not just money.

But lets look later in the same lecture, where he has the students play the “two-thirds the average game” (more on that later). He holds up a $5 bill and says that the winner will receive this as a prize. Now – does he do the analysis of “what people want” (payoffs), which he just said was so keenly important? No, he does not. Only 15 minutes after this front-line defense, it goes forward without comment, that obviously the only value for anyone in the game is the money. So, even though we just lectured on how economic theory can handle different values, we immediately thereafter turn around and act out exactly the opposite assumption. Maybe some people want the $5... maybe some want to corrupt the results for their snotty know-it-all classmates... But no, we get it played out right before our eyes, immediately following the defense that “all values can be handled”, that as soon as money comes in the picture, in practice, we dispense of all other values and speak of nothing except for the cash money.

Critique #2: Economic theory presumes “rational players”, where all the people involved knowingly work to their own best interests all the time. Frankly, that's just downright absurd. People are routinely (1) uneducated or uninformed about what's best for themselves, (2) barred from receiving key information by more powerful institutions or interests, (3) obviously non-rational in instances of emotional stress, drug use, mental failures, and modern Christmas purchasing behavior, and (4) proven by cognitive brain science to be unable to correctly gauge simple probabilities and risk-versus-reward.

Now, consider lecture #1, where Professor Polak introduces game payoff matrices, and the idea of avoiding dominated strategies (that is, a strategy where some other available choice always works out better). With exceeding care, he transcribes each “Lesson” along the way onto the board, including this one: “Lesson #1: Do not play a strictly dominated strategy”. Okay, that's a reasonable recommendation.

But about 10 minutes later, he pulls a devious sleight-of-hand. Analyzing another game, he asks what strategy we should play. “Ah,” he says, “Notice that for our opponent strategy A is dominated, so you know they won't play that, they must instead play strategy B, and thus we can respond with strategy C.” Well, no, that reasoning about our opponent (the 2nd logical step here) is completely spurious; it only make sense if our opponent is actually following our lesson #1. But, have they taken a Game Theory class? Do they know about “dominated strategy” theory? Do they actually follow received lessons? None of those things are necessarily (or even likely, I'd argue) true.

In other words, he assumes that all players are equally well-informed and “rational”, which isn't supportable. And, this assumption is kept secret and hidden. It would even be one thing if Professor Polak came out and said “For the rest of our lectures, let's also assume that our opponents are following the same lessons we are,” but no, he quite scrupulously avoids calling attention to the key logical gap.

And he does is it again, even more outrageously, in Session #2, when analyzing the class' play of the “two-thirds the average game” (a group of people all guess a number from 1-100; take the average; the winner is whoever guessed 2/3 of that average). He has a spreadsheet of everyone's guesses in front of him. Speaking of guesses above 67 (2/3 of 100), he says, "These strategies are dominated – We know, from the very first lesson of the class last time, that no one should choose these strategies." Except that, as he points out mere seconds later, several people did play them! (4 people in the class had guesses over 67; this occurs 46 minutes into lecture #2.) Nontheless, he continues: "We've eliminated the possibility that anyone in the room is going to choose a strategy bigger than 67...". But how can you possibly contend that you've “eliminated the possibility” when you have hard data literally in your hand that that's simply not true? Answer: It's the “rational player” requirement of all economic theory, which demonstrably collapses into sand if the logical gap is recognized and/or refuted. This infected logic continues throughout the class; in sessions #3 and #4 he repeats the same goose-step in regard to "best response" (1:10 into lecture #4: "Player 1 has no incentive to play anything different... therefore he will not play anything different."), and so on and so forth.

Essay on Time Management

2009-07-27T13:17:00.000-04:00

Here's a beautiful essay by Paul Graham called "Maker's Schedule, Manager's Schedule":

http://www.paulgraham.com/makersschedule.html

In brief -- Managers work in hour-long blocks through the day; great for meeting people and having a friendly chat. Makers, however (writers, artists, programmers, craftsmen) work in half-day blocks at the minimum. Interfacing the two -- e.g., managers calling an hour-long meeting at some random open slot in their schedule -- cause the makers to completely lose the in-depth concentration on a task they require. Call this "thrashing" or "interrupts" or "exceptions", if you like. This blows away a half or a full day of productive work when it happens.

Great observation, and it rings extremely true in my own experience. One of the reasons I'm so happy to be outside the corporate environment these days.

Jury Selection

2009-06-05T13:03:00.000-04:00

A few years ago I was hauled into jury duty in Boston, and somewhat disturbed to find that it's impossible for me to ever get selected onto a jury. As soon as you respond differently from all other jurors in the room on any question, you're out (in my case, I was the only one to say "no" to the question "do you agree that you have to follow legal directions from the judge?"). Then a few weeks back my girlfriend was hauled into the Brooklyn court building, and was likewise disturbed to discover the exact same thing (in her case, she was the only one in the room to stick to an answer of "no" in response to, "do you know when someone is lying to you?").

I was confused and mystified by this for a while. We put our heads together with my friend Collin, and I think we finally stumbled into an explanation.

The point is this: Everyone wants to avoid a hung jury (that is, a mistrial, forcing the court & lawyers to try the case all over again another time). The way a jury really works behind the scenes in a criminal trial is that you start with some yes-votes and some no-votes, and over the course of a day or so one side simply batters down the resistance of the other (often through insults and intimidation, as witnessed by another friend), until there is finally a unanimous vote. And who could possibly interrupt this process? You guessed it, the rare personality type who is willing to reject the mob mentality and stand out, disagreeing with everyone else in a crowded, public courtroom.

It seemed odd to me that when we disagreed with the rest of the pool like this, both the prosecution & defense got all jumpy with us about it. You would think (from an expected-value analysis) that if you asked a defense attorney the question, "Which would you rather have as a result of a trial: a conviction or a mistrial?", the answer would be "a mistrial" (since there's at least some probability that your client is found innocent in the next trial). But now I'm guessing that this fails to take into account the opportunity-cost to the attorney in their time; possibly they would actually, ultimately prefer the conviction, and be able to move to other more promising cases, rather than re-try a case which apparently is not a good cause in the first place. (This is similar to the well-known disconnect in incentives between a house seller and the broker working on a commission.) They're not making this loudly known, but I now suspect that avoiding a hung jury may be priority #1 for all the lawyers and judges in selecting a jury, even beyond winning the actual case. Therefore, the able-to-disagree-alone-with-a-room-full-of-people personalities have got to go.

For those of you who want to get out of jury duty, I therefore give a simple, completely foolproof and hassle-free procedure. There's absolutely nothing difficult about it and requires no creativity. Simply pick something, anything in the questions and disagree with everyone else, and you will be immediately released. If you're honest, in fact, it's practically impossible not to do this.

Grading On a Curve Sucks

2009-05-24T12:33:00.000-04:00

Grading on a curve sucks -- there, I said it. For some reason whenever I teach statistics I get joking quasi-comments about "do I grade to a bell-curve", at which point my response is a rather intense diatribe against the very notion. To my perspective, mashing data (grades which are not normally-distributed) to some desired different outcome (the normal bull curve) is simply outright fraudulent, and demonstrates a complete lack of understanding of statistical analysis, or what the normal curve should be used for as an analysis tool.

Back in Fall 2006 Thought & Action magazine published an article by Richard W. Francis (Professor Emeritus in Kinesiology, California State Fresno), asserting thar grading on a curve is the only way to properly compute grades (titled in a propagandist fashion, "Common Errors in Calculating Final Grades"). Here's my letter to the editor from that time:

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Dear Editor,

Richard W. Francis proposes a system for standardizing class grading (Thought and Action, Fall 2006, "Common Errors in Calculating Final Grades"). The system takes as its priority the relative class ranking of students, even though I've never seen that utilized for any purpose in any class I've been involved with.

Mr. Francis responds to some criticism of his system effectively grading on a curve. His response is that instructors can "use good judgment and the option to draw the cutoff point for each grade level, as they deem appropriate". In other words, after numbers are crunched at the end of the term, the grade awarded is based on a final, subjective decision by the instructor. Moreover, there is no way to tell students clearly at the start of the term what is required of them to achieve an "A", or any other grade, in the course.

The example presented in the article of a problem in test weighting seems unpersuasive. We are presented with a midterm (100 points, student performance drops off by 10 points each), and a final exam (200 points, student performance drops off by 5 points each). It is presented as an "error" that the class ranking matches the midterm results. But since the relative difference in the midterm is so large (10% difference each step) and the final so small (2.5% difference each step; even scaled double-weight that's only 5% per step) this seems to me like a fair end result.

Take student A in the example, who receives an "A" on the midterm and a "C+" on the final (by the most common letter grade system). In the "erroneous" weighting he receives a final grade of "B", while in the standardized system he has the T-score for a "D+". Clearly the former is the more legitimate reflection of his overall performance.

As an aside, I have a close relation who was denied an "A" grade in professional school due to an instructor grading on the curve. He still complains bitterly about the effect of this one grade on his schooling, now 40 years after the fact. Any subjective or curve-based system for awarding student grades at the end of a term damages the public esteem for our profession.

Daniel R. Collins
Adjunct Lecturer
Kingsborough Community College

Winning Solitaire?

2009-05-23T18:31:00.000-04:00

Okay, I admit it: Sometimes I play Microsoft Solitaire (i.e., "Klondike" Solitaire: draw 3, with 3 re-deals, Vegas scoring). Of course, it's the most widely-played computer game of all time. Occasionally I go on these benders and play it quite a bit for a few days.

Most games are lost, but I can usually eke out a win in about 20-30 minutes of playing. However, just today I probably lost 30+ games in a row over maybe 2 hours. Still no win so far today. I have to be careful, because I get in a habit of quickly hitting "deal" instantly after a loss (my "hit", if you will), and after an extended time by hand starts to go numb and I start making terrible mistakes because my eyesight starts getting all wonky. (Is it fun? No, I feel a vague sense of irritation the whole time I'm playing, until I actually win and can finally close the application. Hopefully.)

So this brings up the question: What percentage of games should you be able to win? Obviously I don't know, but my intuition says around ~20% or so maximum. I'm also entertaining the idea of building a robot solver, improving its play, and seeing what fraction of games it can win. Apparently this an actually outstanding research problem; Professor Yan at MIT wrote that this is in fact “one of the embarrassments of applied mathematics” in 2005.

The other thing is that all of the work done on the problem apparently uses some astoundingly variant definitions for the game. First, the "solvers" that I see are all based on the variant game of "Thoughtful Solitaire", apparently preferred by mathematicians because it gives you full information (i.e., known location of all cards), and are therefore encouraged to spend hours of time considering just a few moves at a time (gads, save me from these frickin' mathematicians like that! Deal with real-world incomplete information, for god's sake!).

Secondly, they use the results from this "Thoughtful Solitaire" (full information, recall; claiming 82% to 91% success rate) simultaneously for the percentage of regular Solitaire games that are "solvable". But this meaning of "solvable" is only a hypothetical solution rate for an all-knowing player; that is, there are many moves during a regular game of Solitaire that lead to dead-ends, that can only be avoided by sheer luck, for the non-omniscient player. If they're careful the researchers correctly call this an "upper bound on the solution rate of regular Solitaire" (and my intuition tells me that it's a very distant bound); if they're really, really sloppy then they use the phrases "odds of winning" and "percent solvable" interchangably (when they're not remotely the same thing).

So currently we're completely in the dark about what the success rate of the best (non-omniscient) player would be in regular Solitaire. I'll still conjecture that it's got to be under 50%.

Expected Values

2009-05-15T12:26:00.001-04:00

I feel like the idea of “expected values” may be the most important practical concept in probability – and yet, sadly, I find that I completely don't have time to discuss it in any of the math classes that I teach. (Nor is it part of the primary “story” of any of the classes, even statistics). Ever semester I re-visit my schedule and try to find a day to cram it into, and realize again that I cannot.

I've found that probability is enormously alien to a surprising number of students. (Just last week I had students in a basic math class fairly howling at the thought that they might be expected to be familiar with standard dice or a deck of cards). Therefore, I find that I actually have to motivate these discussions with an actual physical game, of the most basic simplicity. If I did cover expected values, here's the rudimentary demonstration I'd use:

The Game: Roll one die.
Player A wins $10 if die rolls {1}.
Player B wins $1 if die rolls {2, 3, 4, 5, 6}
Calculate probabilities (P(A) = 1/6, P(B) = 5/6).

Let a student pick A or B to play, roll die 12 times (say), keep tally of money won on board (use I's & X's). Likely player A wins more money.

Expected Value: The “average” amount you win on each roll.
E = X*P (X = prize if you win; P = probability to win)
Calculate expected values.

Ex.: Poker situation.
If you bet $4K, then you have 20% chance to win $30K. Bet or fold? (A: You should bet. E = $30K * 20% = $6K. If you do this 5 times, pay $20K, expect to win once for $30K, profit $10K)

Speaker for the Dead

2009-05-11T11:22:00.000-04:00

I read Orson Scott Card's Speaker for the Dead over the weekend (while on the road with my band). Can I post about science fiction here? I assume so (seeing as the first post started with a quote from a work by Rudy Rucker).

Of course, I loved Ender's Game. This second book is possibly even more emotionally moving in places (and Card seems to have said he considers it to be the more "important" book to him), but there's a number of notable structural flaws that I'm not able to shake off.

First is that it's very much working to set up further sequels; there's a whole number major plot threads left hanging, and you can start detecting that about halfway through the book (furthermore, I see now that both this and Ender's Game were revised from their original format, so as to set up sequels, which takes away from the narrative thrust at the end of each). Second is that there's a central core mystery that the whole book is set up around, and in places people have to be unrealistically tight-lipped to their closest friends so as to prolong the mystery (I got really super-sick of this move from watching Lost). Third is that the central theme seems like a rehash of Ender's Game (you can very much feel Card wrestling with the rationale to the plot of Ender's Game; you can almost hear him musing "why would an alien race feel like killing is socially acceptable or necessary, anyway?", a central premise of the first book, and then constructing this second book so as to have an actual satisfying reason). There's also some obvious clues that the aliens should have been able to pick up when they kill humans (namely the visually obvious results of the "planting", as witnessed at the end of the book), that would have told them it's a good idea to stop doing such a thing, but apparently they miss them entirely.

But fourth is something that bothers me about lots of science fiction. Although the story spans many years, by way of relativistic time travel (over 3 thousand years, actually), technology never changes during that time. Ender can set off on a 22-year space flight, and when he lands, apparently all the exact same technology is in use for communications, video, computer keyboards, record-keeping, spaceflight landing, government, publishing literature, etc.

In fact, I've never seen any science-fiction literature that manages to deal with Moore's Law (the observation that computing power doubles ever 2 years or less). It would be one thing if they conjectured that "Moore's Law ended on date such-and-such because of so-and-so...", but it's always a logical gap that's completely overlooked. Ender is honored to be given an apartment with a holoscreen with "4 times" the resolution of normal screens... but I'm thinking, in 22 years time, the resolution of every screen should be 1,000 times the ones he left behind on his space-flight. At that rate, I wouldn't bother walking into the next room for one with only "4 times" the resolution.

Maybe that's a subject that is simply impossible to treat properly in a work of centuries of science fiction, but the repeated logical gap (in the face of our own monthly dealings with new technologies) is something that's bothering me more and more. Maybe the Singularity will come and solve this problem for us once and for all.

Forecast: Hazy on Probability

2009-04-21T01:33:00.001-04:00

There's this article at LiveScience.com under the headline "Rainy Weather Forecasts Misunderstood by Many" (here). It reports that about half of all people don't understand what the "probability of precipitation" means in a weather forecast.

If, for example, a forecast calls for a 20 percent chance of rain, many people think it means that it will rain over 20 percent of the area covered by the forecast. Others think it will rain for 20 percent of the time, said Susan Joslyn, a cognitive psychologist at the University of Washington who conducted the study.

Of course, how the article should really be titled is simply "Probability is Misunderstood by Many". I see the same -- dare I say stunning -- difficulty that enormous numbers of college students have in interpreting the most basic probability statements. Unfortunately, in the classes that I teach probability is never more than a quick two-week building-block on the way to something else (either in a survey class, fundamentals of inferential statistics, etc.). Part of me wishes we could give a whole semester course in probability (and basic game theory?) to everyone, but I know there's no room for that in the basic curriculum.

Having seen the difficulty, I've tried to emphasize the interpretation process more in later semesters, and tend to run into more and more resistance against it. Even students who are in the habit of happily crunching on formulas and churning out numerical solutions can be vaguely frustrated and unhappy at being asked what the numbers mean.

This is one where I find it really hard to empathize with the students on the issue (that being rare for me), and I almost can't begin to imagine where I need to start if I get an incredulous response as to how I knew that 75% was "a good bet" if I mention that in passing. Perhaps just growing up in an environment where I was personally steeped in games as recreation every day for decades (chess, craps, Monopoly, Risk, poker, D&D -- see here for more) marinated the fundamental idea of probability into my brain in a way that can't be shared in a class lecture.

Anyway, some people have suggested that statistics needs to be taught to everyone functioning in a modern society. Even more fundamental (after all, it's foundational to statistics) would be getting the majority of people to have a sense for probability in their gut, because most people currently do not. I'd hypothesize that psychological experiments like those at U. Washington have promise of cornering the precise way that our brains are fundamentally irrational -- that math so simple could be so bewildering in practice, suggests a deep limitation (or variant prioritization) in our cognitive abilities.

Blogging as Software Development

2009-04-20T12:18:00.000-04:00

A core principal of open-source software development is something like this: "Release early. Release often. And listen to your customers." (See here.) The idea being, it's better to get something out to your users -- even buggy or incomplete -- so as to (a) get first-mover advantage, and (b) find out exactly what your users want fixed or improved first, because it's rarely the same as what you'd expect on your own.

I'm finding that with the advent of electronic publishing/easy blogging, I'm doing the same thing with my writing. I frequently post something and then go back -- hours or days or weeks later -- re-read it, and make minor (but occasionally numerous) changes to the grammar, sentence structure, and so forth. Sometimes I add in a new anecdote, analogy, or quote that I've come up with in the meantime.

Now, once upon a time we all had to do this the other way around. When publishing was entirely by print -- fixed and labor-intensive -- then ideally you'd write, draft, revise, edit, etc., prior to the final "official" version being published and observed by any readers. (Back when I was a high-school student working on an actual typewriter, I would personally skip the draft/revise process, but I'd take a long time mentally picturing each paragraph and sentence before I put it on the page. Excepting that time I was writing a paper in the morning, last minute, with my grandfather sitting on the stairs waiting to drive me to school.) The point being, what would our older teachers think if we told them that we could entirely reverse the process -- publish our rough draft first, and then instantly add any revisions we wanted, while people were reading and responding to what we had written?

I'm finding that it's a lot healthier for me, now that blogging software is widely available, to reverse the process in exactly this way. I get my stuff out in the world and get some kind of feedback almost immediately. I can get in the flow of the writing/thinking process without getting interrupted too much by the need to stop and pull out a dictionary or a thesaurus. I can put the draft out there and only come back to it if I truly have a really good idea to add or modify what I've written later on. It feels almost like publishing was just waiting to be done this way for the entire history of writing.

On Classroom "Contracts"

2009-04-09T23:22:00.000-04:00

What follows is an open letter I sent to the editor of Thought & Action magazine.
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In the Fall 2008 Thought & Action magazine, professor P.M. Forni had an article called "The Civil Classroom in the Age of the Net". Within that article, he recommends a commonly-seen tactic referred to as a "contract" or "covenant" with the students in the class. Professor Forni writes (p. 21):

Read the covenant to your students on the first day of classes and ask them whether they are willing to abide by it. You can certainly make it part of the syllabus, but if you prefer a more memorable option, bring copies on separate sheets. Then, after the students' approval, you will staple the sheets to the syllabi just before distributing them to your class. Either way, it is of utmost importance that you do not change the original stipulations during the course of the term.

Personally, I think this is one of the more corrosive practices that I've seen widely used in colleges these days.

First of all, the practice is morally ambiguous in that it demands agreement to something being called a "contract" without an opportunity for fair negotiation on both sides. If a student actually does not agree to the presented covenant, what then? In truth, the point of negotiation is when the student formally registers for the class. When instructors bully a classroom of students into a signing statement on the first day of class, we're giving a terrible lesson into the gravity and consideration they should take before signing their name to any document.

Secondly, there is a message usually delivered along with these "contracts" along the lines of, "the covenant is an ironclad agreement that can never be broken". That is again a misrepresentation of how contracts are actually used in the business world. Contracts attempt to establish principles of intent, but they are routinely re-negotiated and amended all the time. When a disagreement erupts between parties, the existing contract may be used as a starting point for discussions, but if agreement cannot be reached, then arbitration or a court case may result. If this were not so, then the entire field of contract law would not exist.

Thirdly, the common usage of these so-called "covenants" causes some students in classes where this is not used (such as the classes that I teach) to believe that without a signed contract, they have no behavioral or performance requirements whatsoever. Obviously this is not the case (again, it's really the moment of course registration in which they agree to abide by the professor's classroom policies), but I have seen it argued by students confused by the practice.

The classroom "contract" or "covenant" of behavior is a confusing, frankly deceptive practice, and it should be avoided by conscientious instructors.

Review of "The God Delusion"

2009-03-31T11:11:00.000-04:00

I struggled a bit trying to rationalize posting this on my math-oriented blog. I finally came to the conclusion that (a) the book in question is largely biology science-themed, (b) it regards a subject that does in fact make me pretty angry, and (c) by the end of the review I do touch on topics of probability and statistics. Hence the posting here.

I've read several chapters of Dawkins' "The God Delusion" and I've got to say that it's disappointing. It's a worthwhile project ("consciousness raising" on why it's admirable to be an atheist), one that I've wanted to do myself in the past, but this doesn't quite fit the bill. Mostly it's a matter of style. It's simply to wordy; it's too discursive; it's too English. Dawkins seems unable to go more than a a single page without some lengthy outside quote; it feels like I just barely get into his train of thought before having to repeatedly jump into some other person's anecdote, poem, or metaphor. I used to take pleasure in nonstop tangents and wordplay like this, but I've found that my patience for it has died out.

I need something that's a bit more punchy, personal, and directly to the point. I would prefer a manifesto and we don't get that here. Dawkins clearly demonstrates a great deal of literary and cultural knowledge, but I find it altogether distracting. In addition, the foils that he's primarily skewering generally seem to be a batch of kindly, woolly-headed, liberal English archbishops, which seem like very faint opposition. Apparently one of the most common clerical responses that Dawkins hears is "Well, obviously no one actually believe in a white-bearded old man living in the sky anymore", which seems entirely off-topic to someone such as myself who lives in American society. He feels compelled to say things like "This quote is by Ann Coulter, who my American colleagues assure me is not a fictional character from the Onion," which again, is completely distracting and quizzical to the American reader.

I'll say this: Dawkins has great book titles. "The God Delusion" sounds like exactly something I'd been looking for, perhaps an explanation or theory of exactly why so many people's brains cling to religion. But frankly that's not what you find between the covers. The keystone Chapter 4 is titled, "Why There Almost Certainly is No God". That sounds compelling, and I could almost start sketching the chapter out in my head, using the modern statistical science of hypothesis testing as a model. But unfortunately the entirety of the chapter is taken up by Dawkins cheerleading for why the theory of natural selection is so great. Great it certainly is. But at best this chapter explains why God is unnecessary for the specific purpose of explaining the evolution of species. People use the idea of God for many, many purposes beyond that, and I think that a far more offense-directed argument needs to be made to fulfill the promise of Chapter 4.

Given Dawkins' focus on biological science and evolution, he has a razor-sharp sensitivity to arguments that "Such-and-such an organ is so complicated, it must have been designed by God"; he spends swaths of several chapters fighting them. Okay, that's a reasonable thing to be irritated by, but here's two observations. One is that I can summarize his argument in a single line. The response to any cleric's "What is the probability that organ X or universe Y could have appeared spontaneously?" should always be "Enormously greater than the probability that a sentient, all-knowing, omnipotent, thought-reading, personally attentive, prayer-answering God could have appeared by chance!" There, I just saved you about 3 chapters.

Secondly, I cannot help but take away the impression that we're fundamentally winning against such arguments. Dawkins makes a good point that a "mystery" to a scientist represents the starting point for an intriguing research project; whereas for a religious person it is a stopping point whose dominion must be reserved for God (historically, complete with threats of violence against exploration). But clearly the "God of the gaps" proponents are being pushed further and further back, perhaps even with greater velocity over time. Whereas previously they would point to organs such as an eye or wing as being impossible to evolve (and since having had the opposite be demonstrated), they have now, according to Dawkins examples, retreated to areas such as microbiology and the flagellum of bacteria. Presumably next will be quantum physics, and beyond that, some unidentifiable regress. My point here is that Dawkins' examples seem to take the emergency out of the issue, and at least from his focus on biological science, it seems like there's little we need to do to disprove God except to support ongoing biology research. I suppose that's good news, but I was looking for more of a direct call-to-action.

In Chapter 5 ("The Roots of Religion") Dawkins has some speculation on the question of "Why does religion exist?". To me, I felt like this was very specifically the promise of a book title "The God Delusion". But Dawkins has no specific thesis, he only has a loose collection of a half-dozen tentative speculations. The most tantalizing are the sections called "Religion as a by-product of something else" and "Psychologically primed for religion" (the centerpiece being, maybe children are mentally wired to implicitly trust what their parents say, so as to pass on key survival skills, and that leaves our species vulnerable to mind-viruses such as religion). It's an intriguing section, but it's short, Dawkins doesn't develop it greatly, nor does he stake out a specific position for it. My preference would be for him to have developed a specific, detailed thesis on the subject before presenting it in a book called "The God Delusion".

In summary: A commendable project, a great title, but a disappointing and distracting read for the American reader.

The Oops-Leon Particle

2009-03-20T19:18:00.000-04:00

I think this is a great 3-paragraph story:

http://en.wikipedia.org/wiki/Oops-Leon

In short, in 1976 Fermilab thought it discovered a new particle of matter, but turned out to be a mistake. It was originally called the "upsilon", but after the mistake was caught, it was referred to as the "Oops-Leon", in a pun on the lead researcher, Leon Lederman. I love that wordplay.

The other thing I love is that, like all modern science, the mistake is partly due to statistics, which we must understand as being based on probability. Looking at a spike in some data, it was calculated that there was only a 1-in-50 chance for it not to have been caused by a new particle (that is, a P-value). But with further experimentation it turned out that that was a losing bet; it actually had been some random coincidence that caused the data spike.

That's the kind of thing you need to accept when using inferential statistics; all the statements are fundamentally probabilistic, and some times you're going to lose on those bets (and hence so too with all modern science). Apparently the new standard before publishing new particle discoveries is now 5 standard deviations likelihood, or more 99.9999% likelihood that your claim is correct.

And you know what? Someday that bet will also be wrong. Such is probability; so is statistics; and hence so is science.

PEMDAS: Terminate With Extreme Prejudice

2009-03-06T14:38:00.001-05:00

Wednesday night, I walk into a lecture room for my first evening algebra class of the spring. And what do I see on the chalkboard? Some motherfucker has oh-so-carefully written out the PEMDAS acronym, with each associated word in a column sequence. In fact, that's the only thing he's got on the board after a presumably 2-hour lecture.

So, now it's time for my official AngryMath "Kill the Shit Out of PEMDAS" blog posting.

It's a funny thing, because I'd never heard of the PEMDAS acronym until I started teaching community college math. None of my friends had ever heard of it; artists, writers, engineers, what-have-you, from Maine or Massachusetts or Indiana or France or anywhere. But for some reason these urban schools teach it as a memory-assisted crutch for sort of getting the order of operations about halfway-right (PEMDAS: Parentheses, Exponents, Multiplying, Division, Addition, Subtraction.)

But the problem is, it's only half-right and the other half is just flat-out wrong. Wikipedia puts it like this ( http://en.wikipedia.org/wiki/Order_of_operations ):

In the United States, the acronym PEMDAS... is used as a mnemonic, sometimes expressed as the sentence 'Please Excuse My Dear Aunt Sally' or one of many other variations. Many such acronyms exist in other English speaking countries, where Parentheses may be called Brackets, and Exponentiation may be called Indices or Powers... However, all these mnemonics are misleading if the user is not aware that multiplication and division are of equal precedence, as are addition and subtraction. Using any of the above rules in the order addition first, subtraction afterward would give the wrong answer..."

In my experience, none of the students who learn PEMDAS are aware of the equal-precedence (ties) between the inverse operations of multiplication/division and addition/subtraction. Therefore, they will always get computations wrong when that is at issue. (Maybe prior instructors managed to scrupulously avoid exercises where that cropped up, but I'm not sure how exactly.)

Here's a proper order of operations table for an introductory algebra class. I've taken to repeatedly copying this onto the board almost every night because it's so important, and the PEMDAS has caused so much prior brain damage:

Parentheses
Exponents & Radicals
Multiplication & Division
Addition & Subtraction

Notice that each operation (after parentheses) is linked with its exact opposite (inverse) in equal precedence. Of course, you have to emphasize that (a) "parentheses" means "stuff inside the parentheses", and (b) in each stage you'll go left-to-right in the expression doing either the operation or its inverse in any order. In contrast, PEMDAS gets parentheses right (and that's pretty much it); but among its flaws are (1) leaving out radicals as the inverse of exponents, (2) overlooking that multiplication & division are tied, and (3) overlooking that addition & subtraction are tied.

An example I use in class: Simplify 24/3*2. Correct answer: 16 (24/3*2 = 8*2 = 16, left-to-right). Frequently-seen incorrect answer: 4 (24/3*2 = 24/6 = 4, following the faulty PEMDAS implication that multiplying is always done before division).

If you're looking at PEMDAS and not the properly-linked 4-stage order of operations, you miss out on all of the following skills:

(1) You solve an equation by applying inverse operations (i.e., cleaning up one side until you've isolated a variable). If you don't know what operation inverts (cancels) another, then you'll be out of luck, especially with regards to exponents and radicals. Otherwise known as "the re-balancing trick", or in Arabic, "al-jabr".

(2) Operations on powers all follow a downshift-one-operation shortcut. Examples: (x^2)^3 = x^6 (exp->mul), sqrt(x^6)=x^3 (rad->div), x^2*x^3 = x^5 (mul->add), x^5/x^3 = x^2 (div->sub), 3x^2 +5x^2 = 8x^2 (considering a shift below add/sub to be "no operation"). If you don't see that, then you've got to memorize what looks like an overwhelming tome of miscellaneous exponent rules. (And from experience: No one succeeds in doing so.)

(3) Distribution works with any operation applied to an operation one step below. Examples: (x^2*y^3)^2 = x^4*y^6 (exp across mul), (x^2/y^3)^2 = x^4/y^6 (exp across div), 3(x+y) = 3x+3y (mul across add), sqrt(x^2*y^6) = x*y^3 (rad across mul), etc. However, the following cannot be simplified by distribution and are common traps on tests: (x^3+y^3)^2 (exp across add), sqrt(x^6-y^6) (rad across sub), etc.

(4) All commutative operations are on the left, all non-commutative operations are on the right (the way I draw it). Also, any commutative operation applied to zero results in the identity of the operation immediately below it. Examples: x^0 = 1 (the multiplicative identity), x*0 = 0 (the additive identity), x+0 = x (no operation), etc. The first example is usually forgotten/done wrong by introductory algebra students.

(5) The fact that each inverse operation generates a new set of numbers (somewhat historically speaking). Examples: Start with basic counting (the whole numbers). (a) Subtraction generates negatives (the set of integers). (b) Division generates fractions (the set of rationals). (c) Radicals generate roots (part of the greater set of reals).

(6) Finally, per my good friend John S., perhaps the most important oversight of all is that PEMDAS misses the whole big idea of the order of operations: "More powerful operations are done before less powerful operations". I write that on the board, Day 1, even before I present the basic OOP table. It's not a bunch of random disassociated rules, it's one big idea with pretty obvious after-effects. (See John's MySpace blog.)

So as you can see, PEMDAS is like a plague o'er the land, a band of Vandals burning and pillaging students' cultivated abilities to compute, solve equations, simplify powers, and see connections between different operations and sets of numbers. If you see PEMDAS, consider it armed and dangerous. Shoot to kill.

Never-Ending Amazement

2009-03-06T13:16:00.001-05:00

I started my spring semester's classes in the last few days, including two introductory algebra classes. It's possibly the best and most powerful start to a semester I've ever had; I got an extraordinarily good vibe from all my classes. So that's a good thing.

So here's a quick observation I've said aloud several times. For me, teaching college math is a continual exploration of the things people don't know. If I listen really carefully, I continually discover that the most basic, fundamental ideas imaginable are things that a lot of people in a community college simply never encountered. Whenever I start teaching a class and think "oh, christ, that's so basic it'll bore everyone to tears, skip over that quickly," I discover at some later point that a good portion of people have never heard of it in their life.

That's actually a good thing. It keeps me interested with this ongoing detective work I do to see exactly how far the unknowns go. And it provides the opportunity to share in the never-ending amazement, through the eyes of a student who never saw something really fundamental.

Here's one from this week (which I got to run in each of two introductory algebra classes). We're going to want to simplify expressions, with variables, even if we don't know what the variables are. ("Simplifying Algebra", I write on the board.) To do that, we can use a few tricks based on the overall global structure of numbers and their operations. I'm about to give 3 separate properties of numbers, here's the first. ("Commutative Property", I write on the board.)

Let's think about addition, say we take two numbers, like 4 and 5. 4 plus 5 is what? ("9", everyone says.) Now, if I reverse the order, and do 5 plus 4, I get what? ("9", everyone says.) Same number. Now, do you think that will work with any two numbers? With complete confidence, almost everyone in a room full of 25 people all say at once, "No, absolutely not."

So of course, I'm sort of thunder-struck by this response. Okay, I say, I gave you an example where it does work out, if you say "no" you need to give me an example where it doesn't work out. One student raises his hand and says "if one is positive and one negative". Okay I say, let's check 1 plus -8 ("-7"). Let's check -8 plus 1 ("-7"). So it does work out. Now do you think it works for any two numbers? At this point I get a split-vote, about half "yes" and half "no".

Okay, what else do you think it won't work for? One student raises her hand and says (and I bless her deeply for this), "if it's the same number". Uh, okay, let's check 5 plus 5 ("10"). And if I flip that around I again have 5 plus 5 ("10"). So now do you think it works for any two numbers?

At rather great length I finally get everyone agreeing "yes" to the Commutative Property of Addition. And it's obviously a point that no one in the class had ever realized before, that addition is perfectly symmetric for all types of numbers. You can sort of see a bit of a stunned look on some people's faces that they hadn't realized that before. Isn't that just incredibly amazing?

After that, we get to think about the Commutative Property of Multiplication (pointing out that commutativity does not work for subtraction or division), look at the Associative and Distributive Properties, do some simplifying exercises with association and distribution, and so on and so forth. But the thing I can't get over this week is that something so simple as flipping around an addition and automatically getting the same answer can, all by itself, be an enormous revelation if you listen closely enough.

Interpreting Polls: An AngryMath Open Letter

2009-02-22T12:43:00.000-05:00

Below, I present the very last question from the last test in the statistics class that I teach, from two weeks ago:

An online forum says "Most of our members are not in favor of switching to a new product. We polled 900 members, and only 37% are in favor of switching (margin of error +/-3%)." Orius reads this and responds, "That poll doesn't convince me of anything. This forum has over 74,000 members, so that's only a small fraction of forum members that you polled." Do you agree with Orius' reasoning? Explain why you do or do not agree. Refer to one of our statistical formulas in your explanation.

Here is the best possible answer to that test question:

No, Orius is mistaken. Population size is not a factor in the margin-of-error formula: E = z*σ/√n (z-score from desired confidence level, σ population standard deviation, n sample size).

Now, I make a point to ask a question like this right at the end of my statistics class because it's an enormously common criticism of survey results. It's also enormously flat-out wrong. (In this case, the quotes I used in the test question were copied directly from a discussion thread at gaming site ENWorld from last year).

Two weeks later, I get up on Sunday morning and eat a donut while reading famed technical news site Slashdot. Here's what I get to read in an article summary on the front page:

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. Adobe puts Flash player penetration at 947 million users out of a total 956 million internet-connected devices, but the total number of PCs is based on a forecast made two years ago. What's more, 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. Is it really possible that 99% penetration could have been reached?

Below, I present my open response to this Slashdot summary:

What's more, 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.

That's the single dumbest thing you can say about polling results. I just asked this question on the last test of the statistics class I teach two weeks ago. Neither the population size, nor the sampling fraction (ratio of the population surveyed), are in any way factors in the accuracy of a poll.

Opinion polling margin of error is computed as follows (95% level of confidence): E = 1/sqrt(n) = 1/sqrt(4600) = +/-1%. So from this information alone, the actual percent of Flash users is 95% likely to be somewhere between 98% and 100%. Again, note that population size is not a factor in the formula for margin of error.

As a side note, polling calculations are actually most accurate if you had an infinite population size (that's one of the standard mathematical assumptions in the model). If anything, a complication arises if population size gets too small, at which point a correction formula can be added if the sampling fraction rises over 5% of the population or so.

There might be other legitimate critiques of any poll (like perhaps a biased sampling method). But a small sampling fraction is not one of them. It's about as ignorant a thing as you can say when interpreting poll results (on the order of "the Internet is not a truck").

http://en.wikipedia.org/wiki/Margin_of_error#Effect_of_population_size

Oops: Margins of Error

2009-02-19T11:17:00.000-05:00

Yesterday I posted an AngryMath blog about polling margin of error, asserting that the following claim was invalid: "If Candidates A and B differ by a number less than the margin of error, you can't be sure who is really ahead."

Well, it turns out that was a mistake on my part. As I groggily woke up this morning (the time where most of my best thinking occurs), I realized I'd made a mistake with a hidden assumption that the percentage of people supporting Candidates A and B were independent... when obviously (on reflection) they're not; in the simplest example every vote for A is a vote taken away from B. You can take A's support and directly compute B's support.

So if I do a proper hypothesis test with this understanding (H0: pA = 0.5 versus HA: pA > 0.5), with polling size n=100 and 55% polled support for A (as an example), you get a P-value of P = 0.1587 (significantly higher than the limit of alpha = 0.05 at the 95% confidence level), showing indeed that we cannot reject the null hypothesis.

In short, it turns out that the statement "A and B are within the margin of error, so we can't be sure exactly who is ahead", is actually correct at the same level of confidence as the margin of error was reported. In fact, to extend that result, there will be a window even if A and B are beyond the margin of error where you still can't pass a hypothesis test to conclude who is really ahead. (Visually, intervals formed by the margins-of-error overlap a little bit too much.)

Mea culpa. I removed the erroneous post from yesterday and left this one.

Basic Teaching Motivation

2009-02-13T12:01:00.000-05:00

I'm constantly obsessing about the best, most important thing I can deliver at the very beginning of the very first meeting of any class. In the past I've basically said that "Abstraction: Familiarity and Use" is the single overarching principle that I'm teaching in all my classes (math or CS), and therefore that should be the introductory lecture, in some sense, in every single class. I think now I might need to get a bit more topically specific for each class.

For the intermediate algebra class that I regularly teach (which is truly an enormous challenge for most of the students I get), I'm considering this very short mission statement: "Can you follow rules? (Can you remember them?)"

(Here's how I might develop this:) When I say that, I don't mean to come off as some kind of control freak. There are both Good and Bad rules in the world. You should take a philosophy course or some kind of ethical training to identify for yourself what rules are Good (and effective, and you should dedicate yourself to following), and what rules are Bad (and you should dedicate yourself to challenging and overthrowing).

But this course is specifically about the skill of, when you're handed a Good rule, do you have the capacity to quickly digest it and remember it and follow it? If you can't do that, then you're not allowed to graduate from college. The purpose is twofold: (1) testing and training in following rules in general, and (2) an introduction to mathematical logic in specific. The first is a requirement before you're expected to be given responsibility in any professional environment. The second gives you a platform to understand principles of mathematics, which are usually the best, most effective, and most powerful rules that we know of.

So, if you can't follow rules, or if you simply can't remember them, it will be frankly impossible to pass a course like this, and you'll get trapped into a cycle of taking this course over and over again without success.

(Honestly, as an aside, I think the primary challenge to students in my intermediate algebra course is simply an incapacity to remember things from day to day. I know now that we can literally end one day with a certain exercise, and have everyone able to do it, and start the very next day with the exact same exercise and have half the classroom unable to do it.)

I conclude, as I've expressed previously before, with a possible epitaph:

I want to foster a sense of justice.
A love of following rules that are good.
A love of destroying rules that are bad.

Division By Zero

2009-02-13T10:27:00.000-05:00

Usually in my intermediate algebra classes someone has to ask "why is division by zero undefined"? This is really highly desirable, because we usually want a moment of discussion to fix the fact for the majority of students who either: (a) have never heard of it before, (b) didn't remember it, (c) don't think it makes sense, or (d) flat-out disagree with it.

One problem that occurs to me now is that, when discussing it extemporaneously, I myself tend to forget that there's two very separate and distinct cases involved: dividing any run-of-the-mill number by zero, versus dividing zero by zero. I think the following is a pretty complete proof:

All variables below are in some set U with multiplication.
Define: Zero (0) as, for all x: 0x = 0.
Define: Division (a/b) as the solution of a = bx (that is, a/b=x iff a=bx).
To prove that division by zero is undefined, let's assume the opposite: there exists some number n such that n/0 = x, that is, n = 0x (def. of division), where x is a unique element of U.
Case 1: Say n≠0. Then n = 0x => n=0 (def. of zero). This is a contradiction that shows this case cannot really occur.
Case 2: Say n=0. So 0 = 0x (def. of division), which is true for all x (def. of zero). Thus x is not a unique number, contradicting our initial assumption.

To summarize: If you try to divide any nonzero number by zero, the answer is "Not A Number". However, if you try to divide zero by zero, the answer is "All Numbers Simultaneously".

(Side note: I guess there's an interesting gap in what I just wrote there, that maybe in Case #2 if the set U just has one single element [zero], then actually you can identify x=0, and do in fact have definable division by zero for that one, degenerate case. Hmm, never noticed that.)

Anyway, that's a fairly intricate amount of basic logic for my intermediate "didn't even know about division by zero" algebra students (2 kind-of-exotic definitions, two cases from an implied "or" statement, and doubled proof by contradiction). So I think what I wind up doing, to give them at least some sense of it, is to consider a single numerical example and gloss over the secondary n=0 case:

Say x = 6/0.
So: x*0 = 6 <-- Multiply both side by zero. Violates rule (for all x: x*0 = 0), so x is not any number (NAN). (Of course, normally you can't multiply both sides of an equation by zero. But you could if division by zero was defined [see degenerate case above, for example] and that actually is the assumption here. It's a sneaky proof-by-contradiction, for one numerical case, without calling it out as such.)

Now, I pick the number "6" above for a very specific reason; if that still leaves someone unconvinced, it gives me the option to do the following. Take six bits of chalk (or torn-up pieces of paper, or whatever), put them in my hand, and write 6/2 = ? on the board. Ask one student to take two pieces out, then another student, and another, until they're all gone. How many students can I do to with before the chalk is gone? That's the answer to the division problem: 3 (obviously enough).

Now take back the chalk and write 6/0 = ? on the board. Ask one student to take zero pieces out, and then another, and another. How many students can I do this to before the chalk is gone? That's the answer to this division problem: Infinitely many, which is (again) not a number defined in our real number system. (If we have a short discussion about infinity at this point, and even if I leave students with the message "I want to say this should be infinity", I think that's okay.)

This is all part of an early lecture on basic operations with 1 and 0, serving to (a) remind students who routinely trip up over them, (b) serve as a foundation for simplifying exercises (why you're expected to simplify x*1 but not x+1), and (c) serve as an example for how much more convenient/fast it is to express rules in algebraic notation, versus regular English. It's also around that time that we have to categorize different sets of numbers, sometimes resulting in the question "what's not in the set of Real numbers?", for which I recommend the examples of infinity (∞) and a negative square root (like √(-4)). So, a lot of those ideas segue together.

Mathematicus Fhtagn

2009-02-13T10:10:00.000-05:00

Routinely I wake up from dreams, explaining myself to someone in this fashion: I'm trying to wrap up my master's degree. I am always "wrapping up", in the last year of that program. It's like some part of my brain, bent into a closed loop, perpetually running on a treadmill; a Sisyphean ordeal, unable to finish. That is, in some sense, where my heart always is.

Dice Distributions

2009-02-10T01:20:00.000-05:00

I got a bit obsessed with finding complete pictures for a sequence of sum-of-dice distributions. I finally completed a nice spreadsheet with charts of the distributions of everything from 1d6 to 10d6, here:

http://www.superdan.net/download/DiceSamples.xls

It provides a nice picture of the evolution of the distribution, as more dice are added, into one that (a) more closely matches a normal curve, as per the Central Limit Theorem, and also (b) gets narrower and narrower, as the standard deviation of the dice average falls. I printed out the first page (n=1 to 3) for my statistics class, in an attempt to intuitively anticipate the CLT.

The other nice thing here is that all the numbers come out of a programmed macro function for summed dice frequency (which I picked up from the Wikipedia article on Dice, and I wanted to see implemented in code): F(s,i,k) = sum n=0 to floor((k-i)/s): (-1)^n * choose(i, n) * choose(k-s*n-1, i-1).

http://en.wikipedia.org/wiki/Dice#Probability

Sharp-Edged Dice

2009-02-08T13:52:00.000-05:00

Over on my gaming blog, I wrote a post on how to test for balanced dice (including any common polyhedral dice used in role-playing games, such as 4, 6, 8, 12, and 20-sided dice). Basically it's an application of Pearson's chi-squared test, and I computed a rejection cutoff value for the SSE (sum-squared error) at a 5% significance level, if you're rolling the die in question 5 times per side. I hadn't seen that specifically presented anywhere else, and I really wanted to see it in a simplified format. I won't repeat it here, but it's over in my gaming blog at the link below:

http://deltasdnd.blogspot.com/2009/02/testing-balanced-die.html

As a follow-up to that, I tried to do a little research on what game manufacturers do for quality assurance on their dice. While I didn't find that specific information, I did find something that absolutely warmed and delighted my heart. It's the long-time owner of the dice-company Gamescience, "Colonel" Lou Zocchi, engaging in a 20-minute rant about how crappy his competitors' dice are, because they engage in a sand-blasting process that rounds off all the edges of their dice in an irregular fashion, and therefore generates slightly incorrect probability distributions. (As an aside, this actually does match my own independent test of his style of dice he produced.)

Lou's retiring this year and selling the company, but he's still got the fire, and I'll give him an official AngryMath salute for his rant on dice probabilities. Watch it here:

http://www.gamescience.com/

Sampling Error Abbreviation?

2009-01-30T15:21:00.000-05:00

Is there no commonly-used abbreviation for "sampling error"?

Can't use "SE" because that's the abbreviation for "standard error", which is a totally different thing (namely, the standard deviation of all possible sampling errors).

Damn it.

First-Day Statistics

2009-01-24T22:01:00.000-05:00

Here's a demonstration that I was deliriously happy to cook up for the first day of my current statistics class. I think it worked extremely well when I first used it (actually as the second thing we do, immediately after looking at a professional research journal for its statistical notation).

Sampling a Deck of Cards: Let's act as a scientific researcher, and say that somehow we've encountered a standard deck of cards for the first time, and know practically nothing about it. We'd like to get a general idea of the contents of the deck, and for starters we'll estimate the average value (mean) of all the cards. Unfortunately, our research budget doesn't give us time to inspect the whole deck; we only have time to look at a random sample of just 4 cards.

Now, as an aside, let's cheat a bit and think about the structure of a deck of cards (not that our researcher would know any of this). For our purposes we'll let A=1, numbers 2-10 count face value, J=11, Q=12, K=13. We know that this population has size N=52; if you think about it you can derive that the actual mean is μ=7; and I'll just come out and tell the class that I already calculated the standard deviation as σ=3.74. (Again, our researcher probably wouldn't know any of this in advance.)

So granted that we wouldn't really know what μ is, what we're about to do is take a random sample and construct a standard 95% confidence interval for the most likely values it could be. In our case we'll be taking a sample size n=4, calculating the average (sample mean, here denoted x'), and construct our confidence interval. As a further aside, I'll point out that a 95% confidence level can be simplified into what we call a z-score, approximately z=2.

At this point I shuffle the deck, draw the top 4 cards, and look at them.

We take the values of the four cards and average them (for example, the last time I did this I got cards ranked 7, 3, 5, and 4; sample mean x' = 19/4 = 4.75). Then I explain that constructing a confidence interval usually involves taking our sample statistic and adding/subtracting some margin of error, thus: μ ≈ x'±E (again, x' is the "sample mean"; E is the "margin of error"). Then we turn to the formula card for the course and look up, near the end of the course, the fact that for us E = z*σ/√n. We substitute that into our formula and obtain μ ≈ x'±z*σ/√n.

So at this point we know the value of everything on the right side of the estimation, and substitute it all in and simplify (the sample mean x', z=2, σ=3.74, and n=4, all above). The arithmetic here is pretty simple, in this example:

μ ≈ x' ± z*σ/√n
= 4.75 ± 2*3.74/√4
= 4.75 ± 2*3.74/2
= 4.75 ± 3.74
= 1.01 to 8.49

So, there's our confidence interval in this case (95% CI: 1.01 to 8.49). Our researcher's interpretation of that: "There is a 95% chance that the mean value of the entire deck of cards is somewhere between 1.01 and 8.49". That's a pretty good, concentrated estimation for μ on the part of our researcher. And in this case we can step back and ask the question: Is the population mean value actually captured in this interval? Yes (based on our previous cheat), we do in fact know that μ=7, so our researcher has successfully captured where μ is with a sample of only 4 cards out of an entire deck.

That usually goes over quite well in my introductory statistics class.

Backstage -- The Ways In Which I Am Lying: Look, I'm always happy to dramatically simplify a concept if it gets the idea across (in this case, the overall process of inferential statistics, the ultimate goal of my course, as treated in the very first hour of class). Let's be upfront about what I've done here.

The primary thing that I'm abusing is that this formula for margin-of-error, and hence the confidence interval, is usually only valid if the sampling distribution follows a normal curve. There's two ways to obtain that: either (a) the original population is normally distributed, or (b) the sample size is large, triggering the Central Limit Theorem to turn our sampling distribution normal anyway.

Neither of those conditions apply here. The deck of cards has a uniform distribution, not normal (4 cards each in all the ranks A to K). And obviously our sample size n=4, necessary to make the demonstration digestible in the available time, is not remotely a "large enough" sample size for the CLT. But granted that the deck of cards has a uniform distribution, that does help us in it becoming "normal-like" a bit faster than some wack-ass massively skewed population, so the example is still going to work out for us most of the time (see more below).

At the same time, ironically enough, I also have too large of a sample size, in terms of a proportion to the overall population, for the usual margin-of-error formula. Here I'm sampling 4/52 = 7.69% of the population, and if that's more than around 5%, technically we're supposed to use a more complicated formula that corrects for that. Or we could legitimately avoid that if we were sampling with replacement, but we're not doing that, either (re-shuffling the deck after each single card draw is a real drag).

However, even without those technical guarantees, everything does in fact work out for us in this particular example anyway. I wrote a computer program to exhaustively evaluate all the possible samples of size 4 from a deck of cards, and the result is this: What I'm calling a 95% confidence interval above, will actually catch our population mean over 95.7% of the time; so if anything the "cheat" here is that we know the interval has more of a chance of catching μ than we're really admitting.

Some other things that may be obvious are the fact that we're assuming we know the population standard deviation σ in advance, but that's a pretty standard instructional warm-up before dealing with the more realistic case of unknown σ. And of course I've approximated the z-score for a 95% CI as z=2, when more accurately it's z=1.960 -- but you'll notice above that using z=2 magically cancels with the factor √n = √4 = 2 in the denominator of our formula, thus nicely abbreviating the number-crunching.

The other thing that might happen when you run this demonstration is there's a possibility of generating an interval with a negative endpoint (even while catching μ inside), which would be ugly and might warrant some grief from certain students (e.g., if x'=3.5, then the interval is -0.24 to 7.24). Nontheless, the numerical examination shows that there's a 94.8% chance of getting what I'd call a "good result" for the presentation -- both catching μ and avoiding any negative endpoint.

At first I considered a sample size of n=3, which would shorten the card-drawing part of the demonstration; this still results in (numerically exhausted) 95.4% chance to catch μ in the resulting interval. Alternatively, you might consider n=5, which guarantees avoidance of any negatives in the interval. In both those cases you lose the cancellation with the z-score, so there would be more calculator number-crunching involved if you did it that way.

Finally, I know that someone could technically dispute my interpretation of what a confidence interval means above as being incompatible with the frequentist interpretation of probability. But I've decided to emphasize this version in my classes, because it's at least comprehensible to both me and my students. I figure you can call me a Bayesian and we'll call it a day.

More Topology Explaining

2009-01-22T12:36:00.000-05:00

Follow-up to yesterday's post on the standard crappy method of explaining topology:

I was at a presentation about a year ago, where someone tried to explain basic topology concepts to non-mathematicians. Here's they went about it: "Consider a cube of cheese and a donut," they said. "They are different shapes. If you draw a small circle on the surface of the cube of cheese, it can be shrunk down to a point. If you draw a circle on the surface of the donut the right way, it cannot be shrunk down to a point. Strange but true."

I almost fell out of my chair when I heard that explanation.

There's a whole slew of things wrong with explanation: (1) Why a "cube" of cheese? That's only going to serve to confuse people into thinking that the geometric "cube" shape is somehow important to the description, when it's not. Again, the only important thing is that one has a hole and the other doesn't. Use some kind of curved shape to avoid tricking people into thinking that the square-ness has anything to do with what you're explaining. (2) Why "drawing a circle"? Yes, as mathematicians we know that's one way of visualizing the important Poincaré conjecture, but here we have to look at it from the perspective of the non-expert listener. Drawings of things don't shrink and expand, so that only promotes further confusion. Use something from daily life that naturally expands and contracts for your analogy. (3) How the heck would anyone accomplish "drawing on a cube of cheese" in the first place?

Here's how I would explain this.

"Consider an orange and a donut. In topology, the only important difference in their shapes is that one has a hole and the other doesn't. Here's how a mathematician would demonstrate that: With the orange, if you wrap a rubber band around it, you can always flick the rubber band aside so it falls off. With the donut, there's a way to connect a rubber band through the hole-in-the-middle part so there's no way to just flick it off. (You'd have to cut & glue the rubber band back together, but then it would be always hang onto the donut.) Doing this mathematically is one way to detect exactly which shapes have holes in them."

Explaining Topology

2009-01-21T22:06:00.000-05:00

(Revised from a prior commentary):

You know, every time someone gives an elementary description of Topology (a branch of modern mathematics), there's a very standard explanation of it, and I think it's a very, very bad one. They always say something complicated like this (from http://www.sciencenews.org/articles/20071222/bob11.asp ):

Topology studies shapes. Specifically, it studies shapes' properties that are not affected by stretching, moving, twisting, or pulling—anything that doesn't break up the object or fuse some of its parts. The proverbial example is that, to a topologist, a coffee mug is the same as a doughnut. In your imagination, you can squash the mug into a doughnut shape, and it will retain the property of having a hole, namely its handle. A sphere is different. You can stretch a sphere into a stick and bend the stick so its ends touch. But turning that open ring into a doughnut will involve fusing the ends, and that's forbidden.

Huh? What the hell does that mean? You start off saying it's about shapes, then start talking in the negative by saying it's not about a bunch of particular properties of shapes. Then there are two pretty poor examples (asking people to imagine stretching things where bulky parts become very thin pieces; it's unclear what corresponds to what). I've taken a full year in graduate Topology, and sometimes I still have trouble understanding that description. Worst of all is this -- that's not what is really important about Topology studies. No one is ever really interested in stretching anything in a topology course.

Here's what I say in the classes I teach: Topology is the study of connections. That's the real story; it's very simple. Yes, coffee cups and donuts are similar topologically, because they're both connected bodies with one hole through each of them. But topology is really useful for things like the following -- A road engineer categorizes intersections by how many streets meet there. A miniature figure modeller plans how complicated an item they can sculpt, knowing the resulting mold has to stay connected around their figure. A stencil-maker has to make stencils one way for letters that have holes in them, and another way for those that don't (e.g., cut out an "A", "B", or "D" normally from paper and those middle holes get disconnected and fall out; that's not a problem for letters like "C", "E", or "F", which keep the surrounding paper connected.) A subway-rider looks for the easiest route to an evening out on the town, knowing they're restricted to specific connecting trains at specific stations. A traveling salesman wants to plan the fastest, cheapest sales trip between a dozen cities, using available commercial connecting flights; or, my food delivery service wants to do the same thing with intersecting city streets.

These are all Topological problems, dealing with how things are connected (which might be solid shapes, but is even more likely to be cords, knots, network circuits, or car/plane/train paths). I suspect I know why most explainers use the big-complicated-useless explanation, instead of the short-simple-and-effective one -- when categorizing different shapes, mathematicians do utilize functions called "homeomorphisms", which somebody at some point thought was best visualized as "stretching" operations. But, seriously, nobody who's nontechnical is going to care about that technique (no more than say, people care about how completing-the-square is used to develop the quadratic formula).

The point of all that technical work in Topology is, again, pretty simple: How is this shape connected? And hence: Where can I go today with this shape? That should be the focus of our first introductions to Topology, I think, not the damn "stretching" analogy, which is practically a cancer on our attempts to explain the subject.

The AngryMath Manifesto

2009-01-18T23:43:00.000-05:00

"Look at it this way. When I read a math paper it's no different than a musician reading a score. In each case the pleasure comes from the play of patterns, the harmonics and contrasts... The essential thing about mathematics is that it gives esthetic pleasure without coming through the senses." - Rudy Rucker, A New Golden Age

"I find herein a wonderful beauty," he told Pandelume. "This is no science, this is art, where equations fall away to elements like resolving chords, and where always prevails a symmetry either explicit or multiplex, but always of a crystalline serenity." - Jack Vance, The Dying Earth

The preceding dialogues are both from works of fiction. That being said, they may in fact truly represent how the majority of mathematicians experience their work. For example, Rudy Rucker is himself a retired professor of mathematics and computers (as well as a science fiction author). My own instructor of advanced statistics would end every proof with the heartfelt words, "And that's the beauty." I've heard that kind of sentiment a lot.

But I never experienced mathematics that way. I now have a graduate degree in mathematics and statistics, and currently teach full-time as a lecturer of college mathematics, and these kinds of declarations still mystify me. Math has never felt "beautiful" or "poetic". I would never in a million years think to describe math as "pleasurable" or "serene".

Math drives me mad.

My experience of mathematics is this: Math is a battle. It may be necessary, it may be demanding, it may even be heroic. But the existential reality is that if you're doing math, you've got a problem. You very literally have a problem, something that is bringing your useful work to a halt, a problem that needs solving. And personally, I don't like problems; I am not fond of them; I wish they were not there. I want them to be gone, eradicated, and out of my way. I don't like puzzles; I want solutions. And once you have a solution, then you're not doing math anymore. So the process of mathematics is an experience in infuriation.

So, again: Math is a battle. It is a battle that feels like it must be fought. It can feel like a violent addiction; hours and days and nights disappearing into a mental blackness, unable to track the time or bodily needs. Becoming aware again at the very edge of exhaustion, hunger, filth, and collapse.

At worst, math can feel like a horrible life-or-death struggle, clawing messily in the midst of muddy, bloody, poisonous trenches. At best, it may feel like an elegant martial-arts move, managing to use the enemy's weight against itself, to its destruction.

I love seeing a powerful new mathematical theorem. But not because it "gives esthetic pleasure"; I have yet to see that. Rather, because a powerful theorem is the mathematical equivalent to "Nuke it from orbit -- It's the only way to be sure". A compelling philosophy.

On the day that you really need math it will be a high-explosive, demolishing the barrier between you and where you want to go. Is there a pleasure in that? Perhaps, but not from the "play of patterns, the harmonics and contrasts". Rather, it's because blowing up things is cool. Like at a monster-truck rally, crushing cars is cool. Math isn't beautiful or fun for us, but it is powerful, and that's what we need from it.

Of course, I also don't know how to a read a music score, so I'm similarly mystified if that's the operating analogy for most mathematicians. Perhaps I'm missing something essential, but I have to stay true to my own experience. If math is going to be useful or worthwhile then it must literally rock you in some way, relieve an unbearable tension, and change your perception of what is possible.

And so, the battle continues.