Teaching, Training, and Advising for the Harvard Mathematics Department. Oh, and some computer stuff.
I have left Harvard as of July 1, 2008 to take a position at
NYU.
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This month’s Wired contains an article about the Netflix prize, offering $1,000,000 to the person or team that can improve Netflix’s recommendation engine by 10%. It’s an interesting tale of the frontier of data-centric personal services. The “hero” of the article uses not only mathematical algorithms but psychological concepts when gleaning information about past preferences in order to predict future ones.
For instance, there’s the concept of “inertia,” or another way, relativity, when ranking movies on a simple 1-5 scale. Somebody might give the same movie two different rankings depending on the most recent movie he or she watched. If you see, for instance, Gattaca followed by The Matrix, you might give The Matrix a 4 because you think it’s so much better than Gattaca. But if the previous movie was a 5 (I’ll let you fill in your favorite here), well, maybe The Matrix earns only a 3. Mathematically, we would say that the relationship between the set of movies and a person’s ratings for them may not actually define a function as the purely math models assume.
The other cool thing about this article is that it’s written by mathematician Jordan Ellenberg, with whom I went to grad school back in the day (the nineties).
Today I saw in one of the University of Michigan student papers an article about a WSJ blog post about a Chronicle of Higher Education column about a new book about college freshmen and their first experiences away from home. The Chronicle story is called “The Myth of First-Year Enlightenment” and the book tries to burst the bubble of those who picture a monumental eye-opening and metamorphosis during the first year. Instead of undergoing then sudden realization that the world is much bigger than they originally thought and what they had come to value was now clearly so…philistine (this was the word used at my freshman orientation), Tim Clydesdale writes,
Most of the mainstream American teens I spoke with neither liberated themselves intellectually nor broadened themselves socially during their first year out…What teens actually focus on during the first year out is this: daily life management.
Most American teens keep core identities in an ‘identity lockbox’ during their first year out and actively resist efforts to examine their self-understandings through classes or to engage their humanity through institutional efforts such as public lectures, the arts, or social activism. Contemporary teens are practical men and women. They . . . manage their daily lives fairly well. But they are not, by and large, thinking men and women.
Practical rather than paradigmatic shifts, I suppose. Clydesdale writes that some students do undergo big philosophical changes, but those are the exception. They end up becoming college teachers and perpetuate the metamorphosis myth.
How does that affect current college teachers? I think the older I get the more I appreciate the college experience, but I have to remember what I was actually like as a college freshmen. And even then I have to acknowledge that my experiences of learning the discipline I teach aren’t shared by the general population of students taking introductory courses in that discipline. Although Math 160’s at the University of Chicago changed the way I think about mathematics, that doesn’t mean that everyone should take it. But understanding where students do come from helps me reach them.
Getting this away from me: James Lang writes in the Chronicle story that we can do better at motivating our courses with what our students are seeking. No, not grades. But practical-minded students need the tools to communicate, to analyze, to formulate, to critique, to defend, to think, to solve problems, and so on. He says he’s stopped advertising that he plans to change the way students think, leaving it as a covert mission. Now he focuses on teaching them to think in the first place.
A couple of people pointed out this article in USA Today about a mathematics professor who thinks fractions need not be taught in schools.
I know that I work for one of the most famous organizations in the world, and I’m well aware that the things I write on my blog could be misconstrued as coming from my employer or representing positions held by them. So I keep things pretty close to the vest. But I hope I’m not being too controversial when I say that I like fractions and hope they continue to be taught. I’m not quite sure how algebra can be done without fractions (what becomes of rational functions if you can’t abstractly divide polynomials?), and without the skills to algebraically manipulate expressions, calculus becomes very hard to do as well.
Prof. DeTurck compares fractions to roman numerals, which of course were abandoned once the arabic system became more widespread. The arabic system had place value, that is, the idea that the position of a digit within a number changed the value of the number. They also had one of the most important numbers: zero. With these ideas, finitely many symbols can be used to express infinitely many numbers.
The quickest to move to the arabic system from roman numerals were the accountants, and indeed, a big advantage of arabic numerals is the ease of computation. The same could be said of decimals versus factions: it’s usually easier to add decimals. Alexander Hamilton wanted a base ten system for the currency of the United States, rejecting the 12 schillings per pound, 20 pence per schilling system long before the UK themselves did. Computers are built for floating-point arithmetic rather than adding fractions, and so most stock exchanges list decimal prices for commodities now.
But fractions are for more than just arithmetic. And decimals are only useful ways to express numbers when the numbers themselves are expressed as points on a line. But there are other ways that numbers are used. If I show you a pie and offer you 0.25 of it, how would you cut the pie? If you said “I’d cut it in half, then cut one of the halves in half”, then you converted the 0.25 to 1/4 and used the fact that (1/2)(1/2) = 1/4. But if numbers are only lengths, I think the only way to do this is to cut a circular piece out of the center of the pie whose radius is 0.5 of the radius of the whole pie.
And don’t forget that many fractions aren’t expressible exactly in decimals. To get a sixth of the pie (a much more modest slice), you’d have to cut out a circle whose radius is the square root of 0.166666… which is about 0.408248.
Shankar Vendatam reported in the Washington Post a while back about the pyschology of football coaches “going for it” on 4th down.
With just over five minutes to play in yesterday’s game against the New York Jets, the Washington Redskins found themselves on their own 23-yard line facing a fourth and one. The team, which was ahead by just three points, elected to do what teams normally do in such situations: They played it safe and punted rather than try to keep the drive alive.
The Jets promptly came back to kick a field goal, tying the game and sending it into overtime. While this particular story had a happy ending for Washington, which won, 23-20, it illustrated the value of an analysis by David Romer, an economist at the University of California, who has concluded that football teams are far too conservative in play calling in fourth-down situations.
You don’t have to be particularly interested in sports to find Romer’s conclusion intriguing: His hunch about human behavior in general was that although people say they have a certain goal and are willing to do everything they can to achieve it, their actual behavior regularly departs from the optimal path to reach that goal.
[…]
New England Patriots coach Bill Belichick is among those who has said he agrees with Romer, and Belichick happens to be one of the more successful coaches in the league. Two Sundays ago, as the Patriots were piling up an astronomical score against Washington, Belichick took a chance on a fourth-down play and got his team seven points instead of the three he might have gotten had the team tried a field goal.
The article goes on to say that most coaches have ignored Romer’s findings. Last week John Madden made a stir when he gave a game plan for the Eagles for their (then) upcoming game against the Patriots:
“I’m not a big guy for going for it on fourth down — but I think you have to go for it on fourth down [in this game] because you have to get touchdowns instead of field goals. It has to be a very, very aggressive approach because offensively, they’re going to take a very, very aggressive approach to you.”
In the middle of the first quarter of the game, the Eagles did go for it on 4th-and-1 at the New England 15. It was basically their first drive, as the Patriots had intercepted a pass three plays into their actual first drive. Generations of coaches have played it safe in similar situations (early in the game, already down 7), kicked a field goal, and tried again on the next drive. But converting the 4th down led to a touchdown which set the tone for the rest of the game: matching scores until the Patriots scored last and forced another two interceptions.
Cheapano Fibonacci wants you to take advantage of his latest Fabulous Fibonacci Sale. Fares are as low as $1* each way. …$1, $2, $3, $5, $8, $13, $21, and some $34 and $55 fares must be booked on spiritair.com between 12:00 PM ET on November 13, 2007 and 11:59 PM ET on November 14, 2007 for travel on the dates as specified by individual market and by market direction…. Please see the terms and conditions for complete restrictions and details.
Asterisks abound , and you can’t fly any day you want, but the deal is legitimate. For instance, you can fly (if you book today)
from Fort Lauderdale to Nassau for $1
from Fort Lauderdale to Orlando for $2
from Fort Lauderdale to Freeport (Bahamas) for $3
$5 and $8 fares seem to be missing!
from Fort Lauderdale to Grand Cayman for $13
from Boston to Myrtle Beach for $21 (now we’re talking!)
from Atlanta to Las Vegas for $34
from Detroit to Cancun for $55
from Orlando to Agudilla, PR for $89
from Orlando to Kingston, Jamaica for $144
or first class for $233
Even though the Fibonacci sequence grows (asymptotically) exponentially, this is still my favorite airline promotion based on an integer sequence.
Large integers. Really large. A history of the names for large numbers, large numbers appearing in the literature, and a contest to find the largest integer you can fit on a postcard.
Dangerous Knowledge In this one-off documentary, David Malone looks at four brilliant mathematicians - Georg Cantor, Ludwig Boltzmann, Kurt Gödel and AlanTuring- whose genius has profoundly affected us, but which tragically drove them insane and eventually led to them all committing suicide.
“In four slides or less,” the online instructions say, “please provide readers with content that captures who you are.”
Is this a Good Thing or a Bad Thing? Neither, I guess. PowerPoint is a tool, and it can be used for evil as well as good. I hope these applicants read The Cognitive Style of PowerPoint by Edward Tufte. He’s a master of visual expression and has many good points of advice and examples of bad slideshow usage.
The article goes on to say that things may not be as bad as they seem. Applicants are still retaining creativity by submitting poetry, photos, even a play. And the admissions committee says it’s more fun than reading essays.
Bad slideshows aren’t limited to business or government, though. I’ve been to many talks where sitting in the back of the room I couldn’t see the slides because the speaker had crammed so much text onto the page. When that happens, the slideshow becomes distracting.
Instead, I try to put no more than three bullet points on a page. Bullet points shouldn’t be too long; they should only give the next few topics I’m going to talk about. The actual text I want to say goes in a separate document–my notes–which I keep with me (PowerPoint does this for you in the notes pane). If I need to show my audience something more complicated, I put it in a third document–a handout–which I copy and hand out.
I use the excellent beamer class for LaTeX to produce slideshows for my classes. My students love them–it’s a nice resource to have to supplement class notes. I can’t use PowerPoint because of all the equations and mathematical notation I need to put into my slides (don’t you dare say “Equation Editor”), but with beamer I get a nice PDF that I can page through. Hyperlinks in the PDF allow me to jump back and forth, and I can even embed media (not that I have a lot of other media).
The 2007 NBA Draft Lottery was last night and has a nice little probability problem in it. Although it’s pretty painful if you’re a Grizzlies or Celtics fan.
Most professional leagues have a policy of awarding the top picks to the worst teams. But the NBA wanted to discourage tanking a season to get a top pick, and so removed this certainty by instituting a draft lottery to award the top three picks. Teams with worse records are weighted to have a higher probability of winning, but the weights are small enough to make the top pick to the worst team far from certain:
Here’s what actually happened, along with the probability of winning for each team:
As you can see, none of the teams with the worst three records (Memphis, Boston, and Milwaukee), and thus the biggest probabilities of winning, ended up with any of the top three picks. Mathematicians often get the FAQ “What are the odds of that?” Well, let’s see.
Let be the probability that the team with the ith worst record wins the lottery. The second pick is determined by the same rules, except that the team that got the first pick can’t get the second pick. So the probability that team j (i.e., the team with the jth worst record) gets the second pick, given that team i got the first pick is . The third pick is assigned the same way. So the probability that team i gets the first pick, team j gets the second pick, and team k gets the third pick is
All the other teams are assigned picks according to worst record among those remaining. So there’s no more probability to be determined.
From the above you can see that the probability of the lottery ending up precisely this way is f(6,5,4) = 0.00068228, or .068%. Very unlikely. But so is the most likely event that the worst teams gets the first pick, the second-worst the second pick, and the third-worst the third pick. That’s f(1,2,3) = 1.88%. The more interesting event is that none of the worst three teams got any of the top picks. The probability of that is the sum
over all triples (i,j,k) of distinct elements from the set 4, 5, 6, …, 14. There are (11)(10)(9)=990 such triples. There might be a nice combinatorial way to get a closed-form expression for this sum, or you can just use Mathematica to do it. I got 0.0405467, or 4.05%. Not that likely, but you would expect it to happen about once every 25 years. The lottery in its current form has been around since 1990.
Rich Zuckerman at NBC Sports published a table of the likelihood of all the bottom fourteen teams getting any of the lottery picks. The event that no bottom-three team gets a top-three pick is equivalent to the event that the third-worst team gets the sixth pick (think about that for a while). He’s got the same 4.05% figure in that position.
Is the draft lottery a good idea? The math is there, and those who make the rules should be content with the consequences. If having as “unfair” a distribution of picks as this occuring this often is acceptable to the team management, then they should keep it as it is. But the lottery was instituted to discourage teams from “tanking” (not trying to win games) the rest of the season so as to increase their draft position. I’m not sure it does. In a non-lottery system, the team with the worst record is awarded the first pick. In a lottery system, the team with the worst record is awarded the highest probability of the first pick. Among the available choices, that’s still the most attractive, so bad teams still have the incentive to try to have the worst record. But teams who legitimately “earn” the worst record are no more likely to be rewarded with high draft picks. So this system doesn’t reward the honest, or punish the dishonest, it just makes the crime pay less.
(I said earlier that this is particularly painful to Celtics fans. I live in the Boston area, but I am not a Celtics fan. I do sympathize with them, though. My point of view on the lottery is really only based in mathematics and, I guess, game theory.)
Recently a certain 128-bit integer that was part of the key to decrypting high definition DVDs got out onto the web. The company developing the technology demanded that web sites publishing the number delete it because it consisted of a “circumvention technology” under the Digital Millennium Copyright Act (DCMA). Unfortunately, that toothpaste won’t go back into the tube.
The legal argument is that the company holding the technology copyrights also owns the number. If that holds up, then You Can Own an Integer Too.
I’ll let the lawyers wrangle the case. But here’s mine:
I’ve been traveling a lot lately and waiting in line at the airport. So I was really surprised to see an AMS poster about the mathematics of Airplane Boarding. After chasing the poster to a Wall Street Journal article, I found the home page of the guy who’s been studying it closely, and has saved airlines millions of dollars per year.
Nice job. But I want to get off the plane quicker, too!
But practical-minded students need the tools to communicate, to analyze, to formulate, to critique, to defend, to think, to solve problems, and so on. He says he’s stopped advertising that he plans to change the way students think, leaving it as a covert mission. Now he focuses on teaching them to think in the first place.
be the probability that the team with the ith worst record wins the lottery. The second pick is determined by the same rules, except that the team that got the first pick can’t get the second pick. So the probability that team j (i.e., the team with the jth worst record) gets the second pick, given that team i got the first pick is 
. The third pick is assigned the same way. So the probability that team i gets the first pick, team j gets the second pick, and team k gets the third pick is
