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.
2009-03-31
2009-03-20
The Oops-Leon Particle
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.
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.
2009-03-06
PEMDAS: Terminate With Extreme Prejudice
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 MadMath "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 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:
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.
So, now it's time for my official MadMath "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
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
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.
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.
2009-02-22
Interpreting Polls: A MadMath Open Letter
Below, I present the very last question from the last test in the statistics class that I teach, from two weeks ago:
Here is the best possible answer to that test question:
Below, I present my open response to this Slashdot summary:
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
Finally, a somewhat more direct statement of the same idea:
This Is The Dumbest Goddamn Thing You Can Say About Statistics
2009-02-19
Oops: Margins of Error
Yesterday I posted an MadMath 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.
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.
2009-02-13
Basic Teaching Motivation
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.
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.
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