Happy Tau Day!

Happy Tau Day! Celebrating τ = 2π = 6.28..., which simplifies all the math formulas that include 2π to make one full turn around a circle. At our house, we'll be having tacos, and tequila, and tarts. Check out the Tau Day manifesto, especially the excellent video by Vi Hart, linked at the bottom of the page -- http://tauday.com/

(Coincidentally, it's also the Perfect Day, because 6 and 28 are the only "perfect numbers" that appear on a calendar --  http://en.wikipedia.org/wiki/Perfect_number.)


Edit: A full course of full circles for Tau Day!

Reformatted Writing

A short observation that I sometimes use to my advantage: It helps me greatly if I write something in one format, and then look at it in some reformatted view, before finally distributing it publicly. This tends to dredge up a number of subtle errors or weaknesses which I otherwise can't see. In some sense it gives me "fresh eyes" to really read the draft from a new perspective (as opposed to one's memory blocking reception of what's actually on the page). In this sense we might say that "what you see is what you get" (WYSIWYG) is actually a hindrance instead of a help. A few cases:

• Blogger. Editing blog posts here, one uses a rich-text editor that is different from the final HTML markup on the site. One clicks "preview" to actually see it reformatted as it will appear on the Blogger site. Something that happens occasionally is that I might have a duplicated "the the" (or something) across a line break, such that I don't initially see it; when the paragraphs and lines get moved around in Preview, I can catch this much more easily. More generally, I can pick up on weaknesses in sentence structure and places I can clarify much more easily in the second view perspective.


• Lecture Extracts. For a couple of years now I've been providing one-page review guides to students in all of my math classes. I accomplish this by copy-and-pasting all of the special defined terms and procedures in my lecture notes to the review sheet. When I did this, the surprising side-benefit was that I discovered a lot of those definitions had varying formats, tenses, or parts-of-speech, that looked sort of ridiculous when lined up next to each other -- and then I could go back and fix them throughout my class lecture notes. (Arguably you could say this is bad practice due to data duplication; anytime I make changes in my lectures I have to edit the review sheet in parallel. But it turns out that this is not a burdensome task.)


• Java Code. Just recently I've committed to formatting all my personal coding projects in the javadoc format (special comments that can be parsed out automatically to external documentation). This required a change in style that I thought would be irritating, but was much less painful than I expected, and more than compensated for by the benefits. Again, if I write my code once and then generate the documentation and look at that, I'm finding there's a whole lot of places I can improve on variable names, method names, comments on use, etc. Looking at it from the perspective of a true "outsider", with only the cues someone would start with to theoretically get into my code, gives my end product much greater depth and robustness.

So in summary: Write once, view in a totally different format, and then edit. Results are improved to a surprising degree.

Basic Logic Errors

I constantly wish that students were taught rudimentary logic at an early age (links: one, two, three). Just musing about that today, here are three common stumbling blocks I see in different classes due to not being able to read logical statements properly:

1. "If" Statement. In a basic algebra class, we have the rule "If the base is negative, then even powers are positive, but odd powers are negative". Immediately after that, I'll always have some students incorrectly evaluate something like: −5² = 25 (or worse, 2³ = −8) . Note that the base of the exponent is not negative, but some students overlook the check required for the "if" qualifier.


2. "Or" Statement. In an elementary statistics class, we have the rule "To estimate a population mean, we must have either a normal population or a large sample size." Then when I ask the class "Do we need a normal population?", the entire class will always incorrectly respond with "Yes!" the first time. Of course that's not true; they're overlooking that only one case of the "or" needs to be satisfied -- most commonly by a large sample size. It takes several sessions of quizzing them on that before they are sensitive to the question being asked.


3. "And" Statement. In practically any class, we might have the policy, "To pass this class you need at least a 60% weighted average, and a 60% score on the final exam." This constantly causes confusion and aggravation. Testy "So, the final exam doesn't count?", or "So, only the final exam counts?" are questions that I routinely have to address. Obviously, students are unclear on the fact that each of two requirements must be satisfied for an "and" statement like that.

Multiple Choice Chances

Let's say you have a final-exam assessment that is a multiple-choice test, with 25 questions, each of which has 4 options, and requires a 60% score (15 questions correct) to pass. As one example, consider the uniform CUNY Elementary Algebra Final Exam (link).

How robust is this as an assessment of mastery in the discipline? As a simple model, let's say that any student definitely knows how to answer N types of questions, but is randomly guessing (uniform distribution over 4 options) for the other questions. Obviously this abstracts out the possibility that some students know parts of certain questions and can eliminate certain choices or guess based on the overall "shape" of the question, but it's a reasonable first-degree model. Then the chance to pass the exam for different levels of knowledge is as follows:

Obviously, if a student really "knows" how to answer 15 or more questions, then they will certainly pass this test (omitted from the table). But even if they only know half of the material in the course, then they will probably pass the test (12 questions known: 67% likely to pass). Of students who only ever know about one-third of the basic algebra content, but retake the class 3 times, about half can be expected to pass based on the strength of random guessing on the rest of the test (9 questions known: 19% likely to pass; over 3 attempts chance to pass is 1-(1-0.19)^3 = 1-0.53 = 0.47).

Meta-Research Innovation Centre

Interesting article about Dr. John Ionnidis at Stanford founding the "Meta-Research Innovation Centre" to monitor and combat weak and flawed statistical methods in science research papers, especially medicine. Good luck to him!

http://www.economist.com/news/science-and-technology/21598944-sloppy-researchers-beware-new-institute-has-you-its-sights-metaphysicians

Gears of War

When I was a kid, one of my favorite pastimes was the Avalon Hill wargame Bismarck about fighting ships in World War II (see it reviewed on my gaming site here and here). In junior high school, at some point my English teacher asked me what I wanted to do as a career, and was completely apalled when I said "I want to join the Navy and control the main guns on a battleship". (I think I'd share her dismay if someone told me something like that today.)

Anyway, over at Ars Technica, a wonderful article has been written by Sean Gallagher (former Navy officer and IT editor) on exactly how the fire control systems on those ships did their jobs -- solving 20-variable calculus problems in real-time (accounting for moving, pitching, rolling, recoiling, Cariolis-spinning projectiles on both ends) with shafts and gears, with accuracy that is hard to beat even today with digital computers and GPS-driven rocketry. There are lots of insightful videos about the components and gears used to do input, sums, multiplies and divides, and spinning disks that can do complicated functions like trigonometry and more.

To me, this stuff is completely like crack. Check it out:

http://arstechnica.com/information-technology/2014/03/gears-of-war-when-mechanical-analog-computers-ruled-the-waves/

(Also: Further commentary and links at recently-established news site SoylentNews.)

FiveThirtyEight

Nate Silver (statistician who famously predicted all 50 states voting in the last election) recently expanded his FiveThirtyeight blog to a full-blown "data journalism" site. His first post was a manifesto on data, science, statistics, politics, journalism, and honest storytelling in general. I agree with almost all of his observations here. They guy really knows his stuff and has a fiery passion for his particular mission. Great stuff.

http://fivethirtyeight.com/features/what-the-fox-knows/