teaching backward, calculus, part 1

Some (admittedly limited) experience teaching math and physics in high school has led me to believe that the standard approach to teaching calculus is misguided. The way we typically teach math, the solutions all come first, and then we teach students why those solutions exist. But problems always precede solutions in real life. Why not in the classroom?

Calculus was developed as a solution to a very specific problem: the motion of objects through space. Though its applications range far beyond that problem, that original problem remains by far the best way to approach calculus since everybody already has intuition and experience with moving objects.

In a better world, then, calculus will always be approached from a physical perspective, since everyone already (sort of, at least) understands how things move around. Here's how. [This is, incidentally, what I did in the first day of AP physics class, but as you'll see, it's not terribly complicated, and (hopefully) most anyone can follow it.]

Imagine you're standing around with a stopwatch on a road that conveniently has length measurements posted all along it, and a car drives past you. Your task is to measure how fast the car is going the instant it passes the mark that's right at your feet. How do you do it?

Well, speed is just a measure of how far the car goes in some unit of time, say, a second, so you can just start your watch as the front wheels pass the mark by your feet, and then mark off where the front wheels of the car are when the watch reads exactly one second. (We can ignore the fact that, perceptually, this might actually be a difficult task...imagine you have some helpers or something). Let's say it's gone 10 meters, as marked on the road. Then it's speed is just 10 meters / 1 second=10 meters per second. Right?

Almost. What you've measured is the car's average speed over one whole second. But remember we want to find the speed of the car the instant it passes by your feet. Let's say it passed you by quite slowly but then managed to speed up incredibly quickly and travel 100 m by the time your stopwatch reached the one second mark. You wouldn't conclude that it was going 100 meters per second when it passed you.

So, you say, okay, let's not measure the distance it travels in a whole second after it passes me, as it can speed up, slow down, and do all sorts of crazy things in that time! Let's measure the distance it goes in just a tenth of a second!

This approach will have the same problem, but it's definitely getting us closer to what we want. The car can speed up or slow down in a tenth of a second just as it can speed up or slow down in a whole second, but it can't speed up as much! What you'll end up measuring though, is the average speed of the car over one tenth of a second. That's probably closer to the speed we're looking for.

Okay, so make it a hundredth of a second, or a thousandth! Well, you're getting the idea. No matter how small you make the time interval over which you're measuring, the car will always move some finite distance over that time interval. You can basically think of the speed as the distance you travel in some tiny time interval, divided by the time interval. If the car goes 10 millionths of a meter in 1 millionth of a second, then it's speed is very well approximated by .000001 meters/.0000001 seconds=10 meters per second.

[Now, if you want to be more precise, the above definition of speed doesn't quite cut it (but it's close enough, so you can probably skip this paragraph). Really, you take all these different tiny time intervals, say a thousandth, a millionth, a billionth, and a trillionth of a second, and mark off where the car is after each time interval. You find the average speed associated with each time interval as we did above for one second and one tenth of a second. If they're the same, great! You're done. That's your speed. But even if they're different, you'll notice that as you make the time interval smaller and smaller and smaller, the speed you calculate will get closer and closer to some value. That's the speed.]

Congratulations, you now more or less understand the idea behind the derivative—one of calculus's two essential ideas! In this case, what we were looking for was the speed. But here's how we found it: we took the change in position (how far the car moves) and divided by the time interval, meanwhile shrinking the time interval so that it was arbitrarily small. In math jargon, this looks like





where x stands for position, t stands for time, and the Greek letter delta means "change in." This, then, we re-define as the derivative of position with respect to time. We solved our problem, and we generalized our solution to a definition, which will be very useful later on!

In the next post, I'll discuss the standard approach to calculus a little more thoroughly.

No, the electoral college is not a good system

Oh come on, obviously you can't think about anything but the election today anyway! You might as well keep reading, even though you probably already agree. Do not be attempted to check nytimes or cnn, as the election results are still not in. And don't worry, Fivethirtyeight still has Obama above 90%.

Anyway, yesterday, courtesy of Sarah (hi Sarah!) I was pointed to this interesting argument in favor of the electoral college (update! see this one from today in Slate, especially point 1 which is basically the same as the previous link). At first it seemed persuasive. But then I realized the entire argument rests upon a basic flaw of sampling and statistics!

Weingarten says that a close election in 1 or 2 states is a manageable disaster, but a close election nationally would be an unmanageable disaster because every vote would be contested, not just every vote in FL, or every vote in OH, or whatever. This is an appealing point—it would be a nightmare if the campaigns were suing for votes all over the country—but it ignores the fact that the likelihood of a close and contestable election in the statistical sense (explained below) decreases sharply with the number of votes cast. A 0.5% margin of victory nationally is equally likely to, but much more robust, than a 0.5% margin of victory in any one state. A more precise formulation of this same idea: if a candidate wins by 0.5% in a single state, it's much more likely that his victory in that state is a result of random vote-counting errors than if the candidate wins the national popular vote by 0.5%.

How much more likely? It depends on the relative size of the state vs. the national population, but the general relationship is that the statistical robustness of a given margin of victory grows like the square root of the sample size. So if a state has 1/100th the voting population of the country as a whole (like, say, CT), then a given margin of victory is equivalent to a national margin of victory that's only 1/10th as large (since 10 is the square root of 100).

The margin of victory of Florida in the 2000 election was about 500 votes out of over 5 million, or less than 0.01% of the total votes cast.  In order for a national victory in the popular vote to be as narrow statistically, it would have be a margin of less than 0.002%, or just 3000 votes out of about 140 million. Although one Presidential election has been this close (1880), it was way back when the population was much smaller, and that election was dubious for lots of other reasons. And no other popular vote result before or since has been anywhere near as questionable. In general, it remains true that the chance of a close election in one or more decisive electoral states is much more likely than the national popular vote being similarly questionable. Therefore a national popular vote is a much more reliable way to arrive at a clear, decisive winner.

Weingarten's other argument is that the electoral college ultimately legitimizes the electoral process by amplifying the margin of victory, since the winner typically wins a much larger fraction of the 538 electoral votes than of the total votes cast. But this contention seems neither desirable, nor true for any election that's close enough for it to really be an issue. Again, think back to the election of 2000. In that year, the election went to Bush by a mere 537 votes! Does that really legitimize the electoral process? No, it makes it seem incredibly arbitrary, because a national popular vote victory will simply never be that close!

Of course, as far as I know, no state has ever been decided that narrowly either, and so it was probably a one-time fluke as well. But the basic point remains: a narrow-enough margin to be dubious in a decisive electoral state is more likely than a narrow-enough margin nationally, because of the much bigger vote sample nationally.

And then there's all those other traditional reasons to dislike the electoral college. But I won't get into that.

Time to call some Ohians!

Teaching Backwards

One of the most difficult things about teaching math and science is avoiding the temptation to teach everything backwards.

The backward approach roughly follows the same standard scheme, as exemplified in every textbook ever (and far too many classrooms): introduce an idea, term, or a definition, and then explain what it means, how it's relevant, and how to do "problems" or answer "questions" using it.

For example, most high school math textbooks come to a chapter titled "trigonometry" or something like that, give definitions of the trigonometric functions (sine, cosine, tangent, etc.), and then proceed to show how useful they are in solving problems involving triangles.

Later in the chapter, in the section titled "the sum and difference formulas," the book gives the sum and difference formulas for sine and cosine, before giving the proofs, and before even stating a relevant question that would require the sum and difference formulas.

Unfortunately, this approach completely eliminates any and all creative insight into the very real problems at hand, and leaves the student with no reason or desire to acquire that insight. It's as if the definitions of the trigonometric functions were handed down by God, followed by a set of problems to solve that require them.

In the real world, of course, everything happens in the opposite order: definitions don't lead to problems. Problems and questions lead to insights, which lead to generalizations, which eventually lead to generalizations and definitions. Someone tries to find the distance between two points knowing their respective distances to a third point and one of the angles. After studying geometry (or even before), most students have an intuitive sense of how to approach this problem, and with some care and coaxing, you can get them to "discover" the law of sines. But if they learn trigonometry from a standard textbook, they'll never even get the chance, because there it is before the problem is given!

Most students can learn to use and manipulate trigonometric functions fine by the backwards method, but its flaws aren't merely aesthetic. If a student has no sense for the scope of a problem he or she is trying to solve, what reason does he or she have to remember the trigonometric functions beyond the next test, or the SATs? For most students, the sum and difference formulas are something to memorize and then forget, rather than a beautiful solution to a seemingly intractable problem.

On the first day of my 4th grade science class, I handed the students a sheet with just three questions: What is science?, Why do we care about/study science?, and How do we study science? Their answers were revealing and kind of depressing (also somewhat hilarious). How do we study science? Why, from science books of course! Why do we study science? So that if we need something to fall back on, we can be science teachers!


These students, like so many, have completely missed the point (so far at least!). Maybe some of them will miss the point either way. But it seems more likely that the things they learn will leave enduring memories if they have to confront the same problems that the people who actually discovered and developed them had to confront. At least that way, they get some sense of what math and science are really about!

Not-Romney for President

All the media back & forths of this Presidential campaign have somewhat obscured a few basic truths about the Republicans and their ideas for governing the country. Yes, we all know Romney has changed his mind on almost every issue, and nobody knows how he would govern. But the narrative and motivation of his candidacy still rest on a few paradoxical ideas about government and the economy.

The first is the "government can't fix your problems, so elect me to fix all your problems" fallacy. It's worth stepping back every once and a while and realize that this makes no sense. The Republicans have been hammering Obama for four years for directing his focus away from "job-creation," while arguing that the government should do less to create jobs. Less government is certainly a coherent ideological position, but not a good way to make jobs in a recession.

The next contradiction: lower taxes will encourage more people to work, which will bring down unemployment. Huh? This wouldn't even make sense if the US economy were somehow lacking for people looking for work. And anyway the problem is precisely the opposite. To the extent that lower taxes would encourage more people to work (which may not even be the case anyway), they would obviously increase unemployment since there aren't enough job openings for all the people looking for jobs anyway.

And lastly, the deficit. People seem to forget that there's a very specific reason to fear government deficits and debt: higher interest rates. Government debt isn't some vague but inherently evil entity that will erode Your Children's Future if not tackled Right Away. Your children will be richer than you! They'll pay back your debt fine!

On the other hand, if people fear the government will become insolvent and therefore unable to back its debts, then they demand higher interest rates, and these higher rates make private investment a less attractive alternative by comparison. That would be bad. But with interest rates on treasury bonds lower than ever, there's just no reason to make an issue of short term deficits.

And don't even get me started on foreign or social policy.

soccer, chance, and attribution, continued

As I argued in an earlier post, football is an inherently probabilistic game. Here I’d like to expand a bit on what I mean, and look at some (preliminary) evidence.

Clearly, everyone realizes that chance and luck play some role in soccer (indeed, in any sport). I’d like to argue that it plays a rather larger, and more specific role than we might think. In particular my hypothesis is that we can predict the distribution of goals in a soccer match and over a number of matches with a fixed-probability model. Imagine a soccer game is like a series of coin tosses of a very, very unfair coin. In each minute of a soccer match, we toss a coin that has about a 1/35 probability of landing on heads. How many times will it land on heads?

My hunch is that that the number of heads you get in this experiment is the same as the number of goals you get in any given soccer match, which (if true) means that in every minute of a soccer match there’s more or less a fixed probability that one or the other team will score.

This isn’t what we expect, or what conventional wisdom would predict. We like to think that in a 0-0 soccer game, the teams just didn’t attack very well, or defended very well, or both, and that in a 4-3 game, the opposite is true. Those teams really came out with attack-minded tactics and didn’t play defensively at all! And they did brilliantly, too! Right?

more thoughts on euro 2012 houghts on euros

I have to admit I succumbed to the hype of Euro 2012. I was excited to watch soccer every day, to watch some of the best teams and the best players on earth. But the tournament has, overall, been a disappointment. Let's admit it: too many of the games that were, on paper, decent match-ups, have been colossal bores. Starting with Germany-Portugal and continuing through to Portugal-Spain today, the games have been low-scoring (lowest since euro 96 overall), defensive-minded, and lacking general excitement. Even a game like Italy-England, which, to be fair, actually had quite a bit of attacking play and lots of chances (mostly for Italy), ended up with no goals at all. For me it's another sign that football, as a game, simply cries out for more goals. It's becoming a game of who can hold on to slim leads, rather than a game of who can attack the most and create the most chances. In all the rule changes I've suggested over the years for soccer, of course, I've never written about the most obvious, most consequential, and least likely to change in the near future: make the goals bigger. But that's for another time.

As for yesterday's semifinal, a few comments. First, Portugal did, in truth, defend brilliantly through the 90 minutes. Spain certainly were not at their best, and seemed to lack a lot of energy, but even a weak Spain team usually dominates possession and creates a lot more than they did. And people tend to look back (as I've mentioned before) when a team gets a clean sheet and claim that they defended well even if they just, in fact, got lucky, but in this case it was no meager stroke of luck. Portugal did what no team in the tournament had done thus far: they defended high up the pitch and denied Spain's defenders time to play the ball out. Teams have tried this against Barcelona, most notably Man. Utd in the 2009 champions league final, or Madrid in various Clasicos, but it usually doesn't work because if you apply high pressure to such a skilled team, you're vulnerable to quick attacks when the team breaks that pressure. But Spain were unable to do that, lacking, most notably, someone quick to run at defenders through midfield. They don't have a Messi, and until late in the game, they didn't even have a Pedro. Iniesta can usually take up this role, but he was unusually subdued.

An offside conundrum

If you've read anything I've written about soccer before, you probably already know that perhaps nothing annoys me more than when a team is falsely penalized for offside. So in a surprising turn of events, I'll be writing today about a new problem: the offside rule as currently interpreted allows for certain plays that should be sanctioned for offside. As you might, expect, two recent examples from Euro 2012 motivate this post: Bendtner's first goal for Denmark against Portgual, and Jesus Navas' goal for Spain against Croatia. The offside rule should be clarified so that these types of goals don't count.

The "Laws of the Game" state that a player is guilty of an offside violation if two conditions are met: 1) He is in an offside position when the ball is last touched by a player on his team, and, 2 "He is involved in active play by [either] interfering with play, interfering with an opponent, or gaining an advantage by being in that position."

Now back to the two goals I linked above. In the first, Bendtner is in an offside position when the ball is initially crossed to Krohn-Deli. He is clearly not in an offside position when Krohn-Deli heads the ball back across to him. On the initial cross, he is neither interfering with play, nor interfering with an opponent. But surely he gains an advantage by being in an offside position at that moment. If he weren't in an offside position, he would be closer to both Pepe and Bruno Alves, who could more easily track him and mark him on the following play. Now it turns out that both Bruno Alves and Pepe are giant ball-watchers, and simply turned their heads and watched as the play unfolded, in this case. However, even if they were decent defenders, they wouldn't have been able to get back mark Bendtner and prevent the goal, precisely because Bendtner was already closer to the goal. Thus, according to the clear and obvious meaning of the words in the offside rule, Bendtner is guilty of offside.

But wait! Since the offside rule is so complicated, FIFA appends a whole section to the "Laws of the Game" clarifying its interpretation. As you can see if you care to look at page 109-110 (that's right) in this PDF, you'll see that precisely this type of play, is deemed "not an offside offence." In fact, the phrase "gaining an advantage by being in that position" is furthermore defined to encompass only two specific situations, namely being an offside position when a teammate makes an effort on goal that rebounds off the goalpost or the goalkeeper.

That's it! Of course, FIFA can define the rules as it wants. But is it really "fair" in some more objective sense to allow these types of plays to proceed and not be offside? Well, by now you probably know what my answer is! To help you see why I think so, take the situation to its logical extreme. Imagine that one striker on an attacking team is camped out in the opposing team's penalty area (cherry-picking, as we say). Surely, the other team doesn't have to mind him when the ball is in the other half, since he's so far offside! That's the whole point of the offside rule, to essentially eliminate that player from relevance! But clearly, as the rule is currently interpreted, the defending team does have to mind the cherry-picker, because of the following possibility (illustrated in the awesome image below). Imagine a long, well-timed through pass is played toward the defending team's corner flag. One attacking player, who was already running toward the corner when the ball was played, chases it, and is tracked by a single defending player. But not all the defending players were running back when the ball was played because they weren't similarly tracking penetrating runs. But what would normally be a defensive situation totally under control, as the player running toward the ball is under pressure even if he gets to the ball first, is now a very worrying situation, because the attacking player can make a simple pass across to his teammate who is now waiting, onside, inside the penalty area, without a defender anywhere in sight! This is analogous to the goals linked above; according to the rules, the play is not offside, but clearly, it should be, because the cherry-picking player in fact compels the defending team to defend him, in a manner completely contrary to the spirit of the offside rule (and, indeed, to the most reasonable interpretation of the language of the rule).

what's the point of the goal-line referee? re: ukraine-england

More thoughts to come as an overall reaction to the conclusion of the group stage of Euro 2012. But for now, the question that should be on everyone's mind (who just watched the final matches in group D): how did the goal-line referee miss that ball going over the line? I'd use this example as more support for my idea that the goal-line referee is essentially a giant misuse of precious refereeing resources. If there are extra referees in a football match, they should help call offsides, since these calls are missed much more often and, though less tangibly so, have a much bigger impact on the final outcome of matches. In addition, as we see here, even having a goal-line referee doesn't guarantee that goal-line calls will even be made correctly! Several talking points here.

First off, the obvious: England fans will be quick to point out that the play should have been ruled offside much earlier on. So maybe it's not such an unjust outcome that it wasn't ruled a goal. Fine. That doesn't interest me so much right now.

More interestingly, I wonder about two things: 1) the positioning of the goal-line referee, (is he really standing in the best position to judge whether the ball has crossed the line? NO) and 2) the training or instructions given to the goal-line referee (was he trained to stand where he was standing? Lamentably, probably. Was he directed as to how certain he should be that a goal has been scored in making the call for a goal? Probably not?)

For the first point. You'll notice if you watch this video or any other that shows the play in question (the link will probably be removed for copyright by now), that the goal line referee is standing right on the goal line, with his line of sight along it. Now some basic trigonometry should enough to convince you that, since the goalpost has finite thickness, standing here will not allow the referee to clearly see the entirety of the ball crossing the line even if it has, because his view of the ball is partially obstructed by the goalpost. In addition, because in order for a goal to be scored the ball has to cross the goal post from his line of sight, there will necessarily be a discontinuity in his view of the ball. Now he sees it, now he doesn't, as it crosses the goal line. This means, even when the ball goes in the goal, it's harder to follow from this view.

So where's a better place to stand? Closer to the fucking goal, and slightly behind the goal line, to have a better view of the ball. Standing right next to goalpost, slightly behind it, would allow the referee to rely on depth perception, instead of merely using line of sight, to see when the ball has crossed the line. This strikes me as a much better system.

As for the second point, how certain should the goal line ref be that a goal has been scored? Certainly 100% is too high a burden. I'd assume that if he was given any instruction by FIFA, it would be of the "beyond a reasonable doubt" variety, but more likely he was left to decide for himself. But of course, any close call, especially one that happens on the goal line so quickly, comes with a large degree of uncertainty. The best criterion to use in this case is a 50/50 judgment. As long as the goal line referee, who by assumption, is the best positioned to make the call, believes there is above a 50% chance that a goal has been scored, he should call a goal. My hunch is that the referee in this case maybe thought it was a goal, but wasn't sure, so refrained from calling it. What a shame! Would've made for a fine finish.

the tournament so far, quick reaction

After having seen 6/8 initial games (missed Croatia-Ireland and England-France, some initial thoughts and predictions:

1. Favorites: Spain are still my favorites to win it, but only marginally, well below 50% probability. My next pick would be (perhaps surprisingly) the Netherlands (you heard it here first!), who, after being the butchers of WC 2010 came out to play this time around. If they can build some confidence and get out of the group of death after the initial defeat, they will be formidable opponents. Technically they looked quite good and created a lot of clear opportunities. Sneijder played a really impressive game in particular. I hope van Persie can find his form again after a woeful start. Though it's just as likely he and his team will do what the great Dutch teams of the past have always done best: play beautiful soccer and choke under pressure. It will be key for them to come out in the next game and not believe what everyone said about their first game, that they played poorly and need to reinvent themselves. If they play the same way, they'll have a great chance of beating Germany and Portugal and winning the group.

 For Spain, well, they can continue to play with 6 midfielders and could even win the tournament that way with odd goals from Silva, Iniesta and Fabregas, but more likely they'll need Torres or Llorente to score at some point. Especially with Pique and Ramos not looking particularly comfortable with each other at the back (never having played together as a central defense pairing), they'll need more than the 1 goal a game that basically won them the World Cup. Not sure if that's gonna happen. On the other hand, Iniesta really stepped up as Spain's most dangerous player...if he keeps playing like that, he could be the player of the tournament and they won't need a striker.

2. Pleasant surprises: Italy held their own against Spain and are my other pick to advance from Group C. Ukraine were also impressive in midfield and attack (though perhaps not so much in defense, as they were quite lucky to escape with the victory in the end). But in a competitive tournament, home support can make all the difference. Russia also looked formidable in their first game, but the Czechs were awfully poor in defense, so I'll be watching their match closely tomorrow.

3. Disappointments: Portugal and Germany both looked lackluster. The Germans could well pull it together, but their midfield play was surprisingly vertical. Muller stuck to the right flank, Podolski to the left, leaving Oezil to be the only real creative force for them. They'll need more, particularly from Muller, to reclaim the exciting form they showed at the World Cup. I don't see the Portuguese advancing, and it'll be another disappointment for Ronaldo in a major tournament.

netherlands-denmark

The Netherlands were upset today by Denmark, despite playing well and creating some great chances. Full credit to Denmark, who took their opportunity, defended well and kept the ball when they needed to, threatening on attack throughout the second half as well. Sometimes you play well and still lose. That's football, right?

It always hurts me a little bit to see a team play well and lose, but what was really frustrating this afternoon was the inane but predictable commentary. Throughout the second half, they couldn't stop saying how poorly the Dutch were playing, how they were so uncreative and couldn't create any chances to score. Which was, simply speaking, false. This is a common mistake that people make watching soccer: they always align their opinions about how well the teams are playing with the scoreline. But the thing about soccer is, those two different aspects of a game can diverge pretty dramatically. In this case, if van Persie had brought his shooting boots, Robben or Huntelaar had converted their chances, the commentators would have been singing Dutch praises, talking about how their tactics were brilliant and how they really had the will to win. If someone had watched the game with the 30 second sequence that included the Danish goal cut out, they would have predicted that the Dutch were the ones that had scored, not the reverse.

In the end, soccer is a game not just of two teams and their quality of play on a given day, but of individual players, individual moments, and chance. People will try to construct ad hoc narratives to fit the results of matches, especially in short tournaments where every result is crucial, but these shallow analyses are an annoying disservice to the teams and players involved.

Euro 2012: it begins

Now that the dust has finally settled over the tragedy of the Champions league semifinal (and final), it's on the Euros and a chance at redemption (for the game of soccer, that is). A few things to note as the tournament gets rolling:

1. The European Championship is probably the best tournament there is, in terms of the actual football being played, given the concentration of high-level games over the course of a single month. Sorry FIFA, but the Euros offer higher quality play, with fewer teams, and all of them just incredibly good. The World Cup is great for spectacle, but the North American, African and Asian confederations still lag behind Europe and South America and dilute the overall quality of play. (The very best matches are at the club level, but are distributed sparsely throughout the season, only when the very best clubs play each other.)

2. The past four years have been the era of Spanish-style attacking football, with Spain winning the last two international tournaments and Barcelona dominating (to a greater or lesser extent) the club level in Europe. After Guardiola's recent retirement, people in the football world have been talking about how he has changed the game for the better by making the attacking, passing, free-flowing style a staple. Though he deserves enormous credit, Barcelona and Spain were already in ascendance before he took over.

One thing that has contributed enormously—but invisibly, given that no one ever talks about it—to this ascendance is the improvement in the enforcement of offsides over the past decade. People say that soccer needs video review technology. Well, video review has already made an extreme difference, just not in the way people thought it would. Several years ago I watched the 1974 World Cup final, a classic match between Germany and the Netherlands. I was shocked to find that, in those days, players were called offside on plays that, today, most people would recognize as clearly onside. But nobody could rewind the tape in those days and see what was really going on: defending players stepping up after a ball was passed to an attacker, thereby making the play look offside when in fact it wasn't. I was lucky enough to see, in the age of the emerging DVR, how often assistant referees wrongly called offsides because of the the time it took to shift their attention from the source of the pass to the player in question, combined with the flash-lag effect.

The improvement in offside calls (or, mostly, non-calls), while vastly underestimated by players and commentators alike, makes good timing and fast attacking play a much more rewarding tactical option. In possibly the greatest soccer performance of all time, Barcelona's 2010 5-0 massacre of Madrid, 3 of Barcelona's goals were barely onside. Ten or even five years ago, all or at least a couple of them would have been called back before the players even had a chance to finish, and the game would have been much different.

In both of today's matches, crucial non-offside calls played a critical role, not in the results themselves, but in the play of the match overall. In Poland-Greece, the pass that led to Greece's penalty and Scezny's sending off was a very close call, as was the Czech Republic's goal against Russia. These aren't just isolated incidents. They happen game after game after game, and they are slowly having less of a negative impact on the game.

Game on!

Big game today!!

On the eve of the crucial Champion's League semifinal second-leg, I found myself wondering last night about how much professional sporting events actually matter, for me personally, and for others more generally. Perhaps because it's an interesting question. Perhaps because my anxiety about the game today is extremely high! Will my life be awesome if Barcelona beat Chelsea? Will it really be terrible if they lose?

After all, who cares? My team wins, my team loses, life goes on, right? Right?! Does it really make a difference to my day-to-day existence whether Barcelona win the Champions League, at all? Or to anyone's, besides the players and managers, people close to them, and the people who live in the actual cities involved?

A lot of research in positive psychology suggests that people are pretty terrible at predicting what will bring them happiness in the future. Sure, I think the game today is important, but will I really care tomorrow, the next week, or the next month, if the result goes my way? Or will it be like buying an expensive car or winning the lottery, where the resultant happiness might diminish quickly and never really live up to my expectations at all?

The question calls for a bit of reflection about how past sporting events. Have they really provided any lasting happiness, or even, somewhat scarily, become a real part of my identity? My initial inclination is to answer with an emphatic yes, both for single events (Iniesta's goal in 2009, anyone?) and the longer arcs of team performances, like DC United in the late 90's or US national team in 2002. And the disappointments are salient as well, like the 1998 World Cup. How can they really feel so important, so many years later?

The answers have serious implications for human psychology and biology. Loyalties to sports teams are evidence of our clear desire to segment into groups and reinforce the differences between groups. This is most obvious in our support for national sports teams, where supporting one's country is a secondary manifestation of national identity. Support for local sports teams makes sense for similar reasons. But in the era of globalization, more and more people identify with teams on the other side of the globe, and that's where the real puzzle lies. Sometimes the choice of loyalties is arbitrary, sometimes it's not. But once we've chosen sides, we rarely change, and we seek evidence to confirm our choice in moral terms, however tenuous or dubious.

One time one of the players on my own team took this sentiment a bit too far, urging me and our fellow-teammates to focus on and intensify our dislike for the individual members of the opposing team for extra motivation. In context, this seemed reasonable: the other team really did seem like an unpleasant bunch of people. But, any one of us could have easily been on the other team (they weren't much different in background from us, and even if they were, would that make it any different?), and if we had been, we would have been saying the same thing about the players who we were, in fact, playing with and liked. Our sentiments weren't completely arbitrary in this case. It made sense to be loyal to the teammates we knew well and had forged strong bonds with; but at the same time, the circumstances for forming separate teams was largely arbitrary, a mere consequence of where we decided to go to school.

As I've argued before, outcomes in soccer games also contribute to my personal narrative about the game itself, and I'm sure this is true for many other players and fans as well, even neutrals. I really, really want Barcelona to win because their winning helps reduce cognitive dissonance I feel about the game of soccer itself. It makes the game seem relevant not only as a competition but as an art, and helps fuel my hope that it will continue to be a worthwhile artistic endeavor. For me, this is often the most important factor, and Barcelona may be the team, throughout its history, that contributes most to this narrative of soccer as a sport with artistic meaning.

But luckily, the game of soccer is unlikely to change significantly as a result of today's match. When all is said and done, Barcelona will still four of the best attacking players on the planet in Fabregas, Iniesta, Xavi, and of course, Messi. Three of them are younger than 28, and they seem unlikely to leave Barca anytime soon. Whatever happens today, they'll be back next year!!

life and tragic death of recorded music, part 2

Recently I read this lovely article (subscription required, unfortunately) by pianist Jeremy Denk in (but what else?) the New Yorker, and it reminded me to follow up on this post about recordings and music. In the article, Denk describes how the process of recording is inherently stressful and tortured, more so than performance, because of the finality of the product, as well as the expectation and possibility of perfection (in some sense, at least!).

A while after writing that last post I realized a somewhat surprising fact about the recorded music that I love and cherish: it is, almost without exception, all recorded before 1985.

math

For about half of a (school) year now, I've been teaching math at a private school in CT. How to teach math is kind of a hot topic, what with the school reform movement and the lagging achievement by Americans in the sciences. So far, I've struggled to accommodate, or consolidate, two basic philosophies about how children should learn math. It's made for quite a tumultuous experience! I'd say, on average, I spontaneously re-think the lower school math curriculum at least on a monthly basis.

On the one hand, there's the Tiger-mother-inspired, "achievement"-oriented approach to math instruction, which is based largely on the theory that kids—for the most part—learn unconsciously, and need to practice something in order to acquire a certain skill level. On this theory, math is something not to be too much enjoyed, but to be drilled and internalized. It is, in another words, a set of basic skills involving numbers and computations that, once practiced and learned, will set a solid foundation for future quantitative endeavors.

On the other side of the coin, we have the idealist philosophy on math instruction.

authenticity and performance practice, part 1?

To follow up on the earlier post about my recital.....Here are some thoughts expanded from this short introduction I gave before playing Mozart, which you can see here.

Authenticity is a problematic concept when it comes to all art. In the world of classical music, the problem is especially important to address because most of what we do as musicians is reproduce the music of others. We don't typically own, at least in a strictly philosophical sense, intellectual property over our performances, since they're written by someone else. Unfortunately, as the classical music scene struggles with evolving standards of intellectual property involving the internet and duplication, it simultaneously struggles to evolve out of an antiquated and damaging mindset involving all that music in the public domain!

It's one of those things that, if you operate outside the world of classical musical performance, you'll probably say either "huh?" or "who cares?!", and maybe you're not totally wrong. But maybe you would care more if the world of classical music performance weren't so bogged down in its silly performance practices!

Before writing anything more witty about it, I feel I have to lay down the groundwork for what I want to say.