Archive for February, 2010

February 8, 2010

Here’s what happens when you didn’t get any all of your grading done this weekend (but the Saints won!!!!!!! Yay New Orleans!), and then instead of catching up before class your department gathers informally and starts to talk about Friday’s wacky trillion point grading scheme, and if there’s a class that lends itself to that, and someone mentions that Physics would really be the ideal course for this, because they have to deal with measures of scale so often.

Well, there’s only one place for this conversation to go:  the other extreme.  What if you had a class that, instead of having 500 points for the semester, or even 1,000,000,000,000, had only 1 point.  Total.  Exams could be worth 2/10 of a point, each homework assignment might only be worth 1/250 of a point, etc.  If someone had an unexcused absence, you’d knock 1/500 off a grade.   It might make it easier to defend points deducted, too, because you could say, “Look, I only took off 5 thousandths of a point for that mistake, so it’s hardly worth arguing about.”

Plus, this could be a learning experience because you could use Official Prefixes:  “This exam is worth 2 decipoints (not to be confused with decapoints)”.   The trouble is, most science students are pretty familiar with deci, centi, and milli so if you really want to make it…memorable….having 1 point for the whole semester is still too much.

The problem is, in looking online, we don’t really seem to have enough prefixes.  The prefixes start off simply enough (deci, centi, and milli being 1/10, 1/100, and 1/1000 respectively) but then decrease by factors of 1000:  the next smallest amount would be a micropoint, which is actually a thousandth of a millipoint, or 1/1,000,000 of an actual point.     Then there are nanopoints (1/1000 of a micropoint) and picopoints (1/1000 of a nanopoint) and while those words are fun to say, we’d need something closer in scale to a micropoint to be able to distinguish amounts, or else it becomes essentially a 1000 point grading scheme with a twist:  “This exam is worth 200 yoctopoints out of a total of 1 zeptopoint for the semester.”

I guess maybe this isn’t so practical after all.  And that’s truly a shame, because who wouldn’t want to write a grading scheme that used yoctopoints?

What has 8 legs and no room for anything else?  A yoctopus!

Carnivals and Clowns. Clowns who shouldn’t be allowed to grade.

February 5, 2010

The Carnival of Mathematics #62 is up today at The Endeavor, and I can tell you right now that I’m totally jealous of the giant Dorito Sierpinski.    Now I’m looking forward to seeing Nerd High!

Speaking of Carnivals, though wonder of wonder we’re posting about #62 on the day it appears, there was also a Math  Teachers at Play #22 up at math hombre [hey, author John works with a friend of mine!  Yup, the math world is getting smaller by the second.]

So that’s the carnival news.  And clowns, you ask?  Well, I’m thinking that the clowns are the faculty of  my department, for  coming up with a grading scheme that’s so absurd I’m really tempted to use it in one of my classes next year.

Here’s the idea:  Suppose you teach a course and you want to have 3 exams each worth 20% of your grade, homework worth 10%, and a final worth 30%.  One way to do this is to set the midterms at 100 points each, the final at 150 points, and homework scaled to 50 points for a total of 500 points in the class.

So far so good, right?  The problem with this is that if you offer extra credit you have to be careful not to give too much — you wouldn’t award 10 points for being the first to speak in class, right?  (OK, you might, but that would be pretty generous.)  So if you want to be able to offer smaller amounts but not have them sound small, you need to have a larger total number of points.

How large?  How about 1 trillion points!  That ties in nicely to the scale of the national debt, which you can tie together with mathematical literacy and/or an interdisciplinary math/political science activity.   Tests are now worth 200 billion points.  The final is 300 billion.  And now, if a student gives a good answer in class you can off the cuff award them one million points of extra credit!  The student feels good — who doesn’t like to receive a cool million points in extra credit? — and you don’t even have to bother remembering to enter it in your gradebook.  On the other hand, if there are little errors on an exam that you want to point out but don’t necessarily want to penalize (forgetting to write parenthesis, for example, so that 2·(3x+5) is written as 2 · 3x+5 ) you could take off 50 million points.  That’s enough to get anyone’s attention.

I think in some of our classes this would be intimidating, so it’s probably not the best scheme in general.  But in other classes, especially the upper level ones, I think our majors would see this as amusing and, perhaps, a help in internalizing the scale of some of these numbers.

Like I said, tempting.

Things that equal e

February 4, 2010

e-day is coming up on Sunday, and I’ve already started making goodies to share on Friday (not wanting to fall into the trap of burning everything again).  Instead of writing “e” on the top, I’m thinking of putting in one of the following:

• $\displaystyle\lim_{n \to \infty} \left( 1 + \frac{1}{n} \right) ^n$
• $\displaystyle\lim_{n \to \infty} \frac{n}{\sqrt[n]{n!}}$
• $\displaystyle 1+1+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...$
• The x-value for which $\displaystyle x^\frac{1}{x}$ is as large as possible.
• $\cosh{1}+\sinh{1}$
• $\cos{i}-i\sin{i}$
• $\displaystyle \frac{\sinh{\pi}}{\pi}+\frac{2\sinh{\pi}}{\pi}\cdot\sum_{n=1}^{\infty}\frac{(-1)^n}{1+n^2}\left(\cos{n}-n\sin{n} \right)$
• (from OEIS A001113) $\displaystyle \left(\frac{16}{31}\cdot\left(\sum_{n=1}^{\infty}\left(\frac{n^3+n+2}{2^{n+1}n!} \right)+1\right)\right)^2$
• (also from OEIS A003417) The number represented by the cool looking continued fraction [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, …]
$1+1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{4+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{6+\cfrac{1}{1+...}}}}}}}}}$

(Any other good ones?)

What transpired at the JMM

February 3, 2010

Last month we skipped the first week of classes and high-tailed it for the west coast, to the Joint Mathematics Meetings in San Francisco.  The entire city was abuzz with excitement over this event, as evidenced by the San Francisco International Airport which had an entire ceiling thing devoted to mathematical symbols:

And the meetings were fun, which they always are.  I even got my very own sketch of a Brown Sharpie scaring Godzilla, thanks to Courtney (in response to my lovely rendition of Godzilla threatening to eat said sharpie).

As it happened, the last day was the one that turned out to be the most full of surprises.  And the best one — better even than my last minute “Heylet’sgoseeWickedit’srightdownthestreet!”, which we did, but with muted enjoyment because we’d already checked out of the hotel and had all of our luggage which we had to cram into our seats under our feet and whoa! those theater seats are place pretty close together, with no consideration for people who might bring all their worldly belongings to the theater — happened that last afternoon of the meetings, after the exhibits had shut down and the message board area was starting to get that Morning After look.

What happened is that I went to the tables to hang out for a bit,  and saw my friend Karrolyne at a table.  She and I were roommates in Project NExT way back when, so I sat down and we chatted and the conversation turned to kids.  There was another woman at the table that neither of us knew and she joined in the conversation.   The three of us continued to chat about our different and common experiences, and at one point Mystery Mathematician commented about talking to people she’d never see again, and Karrolyne laughed and said to watch out, because you never know!  The world is small.

A little later, when we’d turned to other topics, I gave Mystery Mathematician my email address so she could send me some math info, and she exclaimed that she recognized the address.  Hmmmm.  Very strange — we didn’t remember meeting before.  Then Karrolyne had to leave and while I was saying goodbye to her, Mystery Mathematician figured out how we knew each other.  It was from HERE!  She jumped up smiling and said “I’m Math Mama Writes!”  Sure enough, this person I’d just been hanging out with was none other than Sue VanHattum, and in fact we’d exchanged some emails recently (hence the recognition of my address) but had never met in person.  Indeed, she didn’t even recognize my name, nor TwoPi [who stopped by the table briefly while we were talking], because we’re all mysterious with the alias’s here [I was “heather360” for about a day when Batman and I started the blog, and then got bored with that and switched to Ξ]. Of course, that raises the question of why I didn’t recognize hers, but I can only guess that it’s because subconsciously I figured she lived in my computer screen in New York.

All in all it was a fantastic surprise.  And though we didn’t have a lot longer to talk — as it was I made it to TwoPi’s talk with about 20 seconds to spare — it was a delight to know that she was just as fun to talk to in real life as I would expect from her blog.

Small world, indeed.

A=B implies that 1=1, therefore…

February 2, 2010

I’ve ranted in the past about the fallacy of trying to prove an identity by starting with the equation itself, then manipulating both sides of the equation until you arrive at a valid identity.

While grading some homework this term (involving proofs of trig identities), I found the need to raise the subject again in class.  But my stock example, proving that -3 = 3 by squaring both sides, seemed too transparent.  I wanted something where the fallacy was solely due to proving that False implies True.

I ended up with the following example, which I like a lot, but which I’m certain has been rediscovered by others over the ages.  Still, it’s a good illustration of why we can’t prove identities in this way.

Claim: $\sin x = \cos x$, for all $x$.

Proof: Assume that $\sin x = \cos x$.  If we square both sides, this implies that $\sin^2 x = \cos^2 x$.  Furthermore, since equality is reflexive symmetric, it follows that $\cos^2 x = \sin^2 x$.

Finally, adding these two equations gives us $\sin^2 x + \cos^2 x = \cos^2 x + \sin^2 x$, which reduces to the equation 1=1.  QED.

The careful reader will note that squaring both sides is irrelevant, as is the Pythagorean identity for sine and cosine.  In essence we have a general proof that $A = B$ for any expressions $A$ and $B$:  If $A = B$, then by reflexivity symmetry we know that $B=A$, and thus $A + B = B + A$.  But since addition is commutative, this reduces to the identity $A+B = A+B$.

I prefer the slight obfuscation of the $\sin^2 x + \cos^2 x$ proof over the distilled simplicity of “reflexive symmetric plus commutative”.