Math in the Movies: Stand and Deliver

During the course of my K-12 math education, I was able to watch Stand and Deliver two times during math class. The first time was in 5th or 6th grade, and during this first viewing I was less inspired by the mathematics than by the stellar performance of Lou Diamond Phillips, whose winning catch phrase “I strangled him, his body’s decomposing in my locker” has stuck with me well into my adult life.


The second time I saw the film was in high school, during the month between the AP exams and summer vacation when teachers are generally a little less rigorous with their lesson plans. Wiser now, I was able to more fully appreciate the mathematics on display in the film. I understood what it was like to sit down for an AP Test, and while I’ve never had Andy Garcia accuse me of cheating, I think I can imagine what it would feel like. Because of this, I was able to relate to the film on a deeper level.

Recently, I decided to watch this film for a third time, to see how this film compares with other films that involve mathematics. Lou Diamond Phillips was as charming as ever – but how did the math stack up?

For the uninitiated, Stand and Deliver aims to tell the true story of Jaime Escalante, a man who gained a fair amount of press in the 1980’s for developing an extremely successful advanced placement math program at an inner city school in Los Angeles. The film tells the story of the first batch of kids to study Calculus under Escalante’s tutelage, and aims to show that regardless of your background, an understanding of mathematics is not beyond your reach.


Here’s an extremely short trailer for the film. Run, Lou, run!

Most of the time when I discuss the representation of math in films, there are two main things to consider: the portrayal of mathematicians, and the portrayal of mathematics itself. For this film, Mr. Escalante is not a mathematician, however – he is instead a very good math teacher. Nevertheless, being a good math teacher means he must be good at math, and whenever someone who is good at math is presented on film, there is a danger that the character will have certain stereotypical attributes.

Thankfully, this doesn’t seem to be the case here. Let’s take a closer look at some stereotypes of mathematics.

- People who are good at math are socially awkward.

Jaime Escalante is many things in this film, but socially awkward is not one of them. He’s a charismatic dude with a comb over to match. Sure, some of the things he says may not be entirely appropriate for the classroom, but everyone seems to enjoy it. Plus, he has a stable home life with a loving wife and a son – it’s not often that people who are good at math are shown in such drama free households. +1.

- To be good at math, you must be insane.

This movie shows all types of students, from the nerdy girls who work at their dad’s restaurant, to the wannabe gangsters with no aspirations for higher education. Many of the students come from less than ideal family situations. Nevertheless, there is one thing that binds them all together: their ability to learn math.

While some students are stronger than others, there isn’t one among them who is completely lost. They all learn Calculus, despite the skepticism that surrounds them. At one point Mr. Escalante opines that “students will rise to the level of expectation,” and in this case, he is correct. You don’t need to be crazy to be good at math. Having a good teacher, however, certainly helps. +1.

Real Jaime Escalante versus movie Jaime Escalante.

- Mathematics is inherently difficult and complicated, and only gifted people have a hope of doing well.

This is arguably the most harmful stereotype about mathematics. While it’s certainly true that math is difficult, and that there are those who seem to have an innate mathematical ability, it is certainly not the case that every professional mathematician (or those who use math in a technical career) are math savants. More often, they are simply people who had a few good teachers and were motivated to really understand mathematics.

Usually a film’s mathematics perspective is weighted heavily towards the savant end of the scale, which only reinforces stereotypes about people who study mathematics. Thankfully, this film really emphases the latter standard – that even if you aren’t the most naturally gifted when it comes to math, you can still succeed with enough hard work. All the students in the film work extremely hard, even the ones who may be better at math than the others. And aside from a bit involving Andy Garcia who plays a total tool, the students are all rewarded for their hard work, not only with good grades, but with a deeper understanding of math, and greater confidence about their abilities as students. As Mr. Escalante says, “Calculus is not made to be easy – it already is.” +1.

This film is very different from most films that involve math. Math isn’t presented as some mystical oracle that can only be deciphered by the borderline insane. Instead, it is presented as a difficult subject, but one that can be mastered with dedication and practice. It’s no wonder, then, that this film has secured such an enduring spot in the hearts of math teachers nationwide.

For those of you who may not find the film appealing, there’s always the counterpoint offered by South Park. Mr. Cartmanez may not have the heart of Mr. Escalante, but at least he has the comb over. Their teaching styles couldn’t be more different, and yet in their own way, both are successful. Somehow, though, I don’t think math teachers will find the story of Mr. Cartmanez as appropriate for their students.

March 27, 2009 | No Comments »

The Math of March Madness

With the NCAA college basketball tournament now well under way, no doubt many of you are following the games closely, and vying for your teams to make it to that sacred promised land known as the Final Four. Even the President’s caught some of the madness.


When filling out a bracket, of course you would like to predict as many games correctly as possible. No doubt a thorough understanding of the teams can help in this endeavor, as well as a careful analysis of their performance throughout the season. But none of us is perfect, and we are bound to make some incorrect predictions.

Even if you are quite skilled when it comes to picking winners, and can pick correctly 75% of the time, the odds of you selecting the correct winner for each game of the tournament are about 1 in 74,325,939. Roughly speaking, this means that even if the NCAA Tournament took place every day, rather than over the period of a few weeks once a year, it would take over 200,000 years, on average, for you to have a perfect bracket. At the more reasonable estimate of you picking winners only 50% of the time, things get even worse.

This leads us to a natural question: how can we maximize our odds of picking the correct winners? Many of you probably have your own routines that you use to help you decide, but this article from the Scientific American blog has one useful tip: forget about the seeding from the Elite Eight onwards.


Without question, the seeding can help you accurately predict winners in the early rounds. The fact that no #16 seed has ever defeated a #1 seed should give you four winners for free as you are setting up your bracket. As the field of winning teams shrinks, however, so too does the importance of the initial seeding. This point is confirmed by an analysis of Professor Sheldon Jacobson, who, by looking at tournament results over more than 2 decades, concluded that the initial seeding has as much predictive power in the last three rounds of the tournament as flipping a coin, statistically speaking. In other words, it has no predictive power whatsoever.

So, we can conclude that the usefulness of the seeding diminishes as the tournament progresses. But then we must ask: what parameters are useful in determining winners? To answer this question, we can turn to the work of Dr. Joel Sokol, Dr. Paul Kvam, and Dr. George Nemhauser. Together, Dr. Sokol and Dr. Kvam the LRMC (Logistic Regression/Markov Chain) model for predicting winners in the NCAA Tournament.

What makes their system important is that it is significantly better at predicting winners than other widely used methods. The LRMC website highlights three benefits:

  • LRMC is right more often: When LRMC and other NCAA tournament ranking methods disagree, the team LRMC ranks higher wins significantly more often than the other method’s team.
  • LRMC is particularly effective at sorting out the top teams, as measured by the last three rounds of the NCAA tournament, and it is more successful at identifying “surprise” Final Four teams. (We’re not perfect, though–LRMC didn’t expect George Mason’s run in 2006!)
  • LRMC is more effective at picking potential bubble teams; the teams it ranks in the “last-teams-in” bubble range tend to win more games than the teams that other methods rank in that area.

In particular, the the second point suggests that the LRMC method is able to pick up the slack when the significance of the initial seedings begin to wane.

What makes the LRMC method work? You can find a link to their research paper here, but these two points give a pretty good qualitative explanation:

  • When determining the value of home court advantage, LRMC considers how much playing at home helps a team win, rather than how many “points” playing on a home court is worth.
  • Our research shows that very close games are often “toss-ups”; the better team barely wins more than half the time. So, winning a close game shouldn’t be worth as much as winning easily, and losing a close game shouldn’t hurt a team’s ranking as much as losing badly. In fact, a close loss to a top-tier team often shows more about a team’s quality than a blowout win over a weak team; LRMC’s ranking methodology takes this into account.

These points make intuitive sense. In particular, when looking at a team’s history, one should focus not only on the number of wins, but on the score differences. This is a significant part of the LRMC method, and given the relative strength of the method, this should tell you that the score difference seems must be an important parameter.

However you determine your bracket, chances are mathematics can help you improve your odds. While it may be too late to recruit math’s aid this year, it will be waiting for you in the years to come. And should you find that mathematics gives you a leg up among your peers when March Madness rolls around, I’d be happy to accept a portion of any winnings you receive (on math’s behalf, of course). Mathematics needs to eat to, you know.

March 20, 2009 | 1 Comment »

Verizon Employees Suck at Math

If you’ve got the time, and/or the patience, listening to this audio clip of George Vaccaro try to deal with a series of Verizon representatives who claim that 0.002 = 0.00002 should be enough to strike fear into your heart regarding the future of mathematical literacy in this country. Then again, he’s talking about problems he had while in Canada, so maybe the reps are Canadian. We’d never make such an obvious mistake here in the States, right? Right…

On a related note, I would encourage all of you to start writing the dollar amounts on your checks as more complicated mathematical expressions. Everyone could use a boost to their mathematical literacy, bankers included.

The audio clip is quite long, and the longer it goes on, the more depressing it gets.

March 19, 2009 | 2 Comments »

Pi Day

Hot on the heels of Square Root Day comes Pi Day, a day held in honor of arguably the most famous mathematical constant, π. And like Square Root Day, I am forced to approach this holiday with a certain degree of hesitation.

There is no doubt that Pi Day is the most prestigious mathematical holiday, but this recognition usually only serves to illustrate the sad state of mathematical literacy in this country. For example, one year I remember reading a news article about Pi Day where the author described π as a number whose decimal expansion “was believed to go on forever.” Of course, belief has nothing to do with it – this is a simple consequence of the irrationality of π, a fact which is apparently lost amidst the pie eating hubbub of this holiday.


Unfortunately, this is not an isolated incident – for as much as Pi Day aims to educate people about π, it seems to do just as good a job of showing how little people actually know. Searching Google News for articles on the upcoming holiday, it’s possible to find a number of stories that say a whole lot of garbage. For example, there’s this quote from a Pi Day article on SF Gate:

Pi, as [Pi Day co-organizer Ron] Hipschman noted, is strange because it’s both an irrational number (its decimal expansion never ends or repeats) and yet the number is also transcendental (no finite sequence of algebraic functions could ever produce it).

To a physicist like Shaw, that kind of contradiction and beauty was all the inspiration he needed to contemplate a Pi Day.

This sort of writing is like nails on a chalkboard to anyone who knows better. Forgetting the convoluted definition of a transcendental number given above, the more important point is that there is no “contradiction” in the statement. There is nothing special about the fact that π is both transcendental and irrational – as is immediate from the definition of a transcendental number, any transcendental number is automatically irrational.

No doubt there will be other examples of this mathematical butchery as Pi Day draws near. Here are a couple more. From the Times Online:

[S]ince 1988 mathematicians across the land have been celebrating pi day each year by tucking into a feast of [sticky pudding]. The number has obsessed generations of mathematicians for millennia, and not because it’s an excuse to eat pudding.

As I’ve said before, there’s not a working mathematician today (nor can I think of one over the past several hundred years) who has made a career studying the number pi. No mathematician is “obsessed” with this number – although numerologists and Max Cohen certainly may argue otherwise.

Some sources don’t even seem to know what π is. From Jacksonville, FL:

The next big day to celebrate in the math community this year [after Square Root Day] is pi day, March 14th. It represents 3.14 – a common mathematical expression.

And from Montgomery, AL:
On March 14 math lovers can celebrate Pi, the mathmatical [sic] formula used to find the circumference of a circle’s diameter, which is 3.14.

Not only is π defined incorrectly in both of these quotes, but it’s clear that the authors don’t know that π is neither a formula, nor an expression, any more than the number 12 is a formula or an expression.

One could argue that perhaps I am just nitpicking. For the general reader, such details are of no consequence, you may say. Unfortunately, history has shown that misinterpretations of the number π can lead to quite embarrassing consequences. One need look only to the good people of Indiana for proof.

As discussed in the article linked above, around the turn of the last century, a man named Edwin J. Goodwin claimed to have done what mathematicians already knew was impossible: he claimed to be able to square the circle (i.e., he claimed he had found a way to construct a square with the same area as a given circle, using only a compass, straightedge, and a finite number of steps).

Not content to keep the discovery to himself, Dr. Goodwin decided to share his discovery with his fellow countrymen in Indiana:

The stalwart Hoosier determined that the great state of Indiana should be the first to benefit from what he fervently believed to be a “new mathematical truth.” He would allow the state to use his discovery and to put it in the school textbooks free of charge. There would be no need for Indiana to ever pay him any royalties.

On January 18, 1897, after emerging from the House Swamplands and Education Committees (legislatures sometimes work in mysterious ways), Indiana House Bill 246 was introduced to codify Dr. Goodwin’s discovery. Legislators freely admitted they did not understand the jargon-filled bill, although they were certain it had something to do with circles. Of course they passed it unanimously.

While his heart was certainly in the right place, his mathematical rigor was not. His construction relied upon the unfortunate claim that π = 3.2. Thankfully, the error was pointed out before the bill was able to do any damage. The story does go to show, however, that we have a long history of not understanding π.

Eating pie is certainly an activity I can support, but other than that, I’m not really sure of this holiday’s purpose. On the official Pi Day website, for instance, the three questions on the front page up for discussion are: “Why do you like Pi?”, “What are you doing at your school to celebrate Pi Day?”, and “How many digits of Pi have you memorized?” Note that two of these questions actually have nothing to do with the number π, and the one that does deal with π doesn’t ask about any actual mathematics.

If you’re going to delve into this number, at least ask some interesting questions. How about “How can you show that π is irrational?” (here is a simple proof that only requires some basic calculus) or the related question, “How can you show that π is transcendental?” For younger students who may not understand or appreciate such proofs, how about “Where are some unusual places that π appears?” (You could show them the infinite series π/4 = 1 – 1/3 + 1/5 – 1/7…, which is usually quite surprising to a first time viewer). Or, for a more philosophical question, “Why does π appear in so many places in mathematics?”

I’m also a reluctant supporter of this holiday because I don’t really see a reason for π to steal all the limelight from other constants that may not have the PR that π does. There are other constants equally deserving of our attention. This is a slippery slope, of course, and once we say this, it’s natural to say that there are certainly more important concepts in mathematics, each one deserving of its own day to celebrate.

Perhaps in time, we will see more effective use of these “math holidays.” For now, though, I think that this is about the best we can expect to get:

March 11, 2009 | 1 Comment »

Optimization at the Checkout

On more than one occasion, while waiting in line to buy my lunch on campus, the cashiers at the front have asked those of us in the line to split into smaller lines – one line for each cashier. This seems to be met with hesitation on the part of those of us who are in line, and rightly so. Perhaps I am simply projecting, but it seems like they all know the same thing I do: that having only one line feed into all the cashiers is the most efficient way to manage a queue.
One would think the cashiers should know this as well, but apparently not. So, if you have ever asked people to form separate lines when waiting to be helped, pay attention, because you need to learn why people in line rarely pay attention to you.
For a person waiting in a single line, there is little incentive to break into smaller lines. This is because using several lines leads to longer wait times on average. You don’t need any sophisticated machinery to explain why this is true – if you ruminate on the two choices for a moment, the benefits of the single line system should make themselves apparent.
With only one line, you never have to worry about getting stuck behind a coupon-clipper or a check-writer. You move forward whenever anyone’s transaction is completed, which means that even though a single line will be longer than several shorter lines, it will also move much faster.

This is also a plus for those of us who have trouble with decision-making. With only one queue, there is no decision to make. You needn’t worry about developing a strategy when picking your checkout line; for example, you don’t have to size up those ahead of you to discern whether or not they are the type who will take a long time paying. Just get in the line and move – it’s really as simple as that.

If only everyone could be as line savvy as Apu, perhaps we would have no need to study queue management.

How much more efficient is the single line queue? Apparently there are tools available that allow to model these sorts of situations, but here is one such example, courtesy of the blog of Dr. Michael Trick:

Suppose you have a single queue with 20 customers arriving per hour. If the cashier can handle (on average) 22 customers per hour (close to saturation, but probably roughly what “efficient” managers would aim for), then the queue will grow so long that the average wait will be 27 minutes! Five such queues would end up with about 50 people waiting in line on average. If you go over to one line (with 100 arrivals/hour) being served by five cashiers, the average wait goes down to under 5 minutes, and the number of people waiting in line is only 12 on average.

This simple example shows that the benefit to a single line is quite significant. So significant, in fact, that many grocery stores are now organizing their checkouts to have a single queue. Whole Foods is perhaps the most prominent example, because of an article the New York times wrote nearly 2 years ago, which came to the unavoidable conclusion that the single line queue is the only way to play.

2 years seems like enough time for such an unequivocal conclusion to have begun seeping into our collective consciousness, but apparently not. I welcome the day when I am no longer asked to form separate lines while buying my lunch – it’s like asking me to give you even more of my (quite valuable) time. It’s not that we can’t hear you, cashier, it’s that we know what’s in our own best interest. And frankly, so should you.

Is there any advantage to using multiple lines? There may be some psychological benefit to having many short lines rather than one longer line, especially for people who, for example, may go to the grocery store only to pick up one or two items. For them, the sight of a single line may be overwhelming, even if that line does move much faster than separate shorter lines would.

Also, there is perhaps something to be said for the use of express lines, which cater to those people who would be most turned off by a long line. However, with the single line system, that one line is already express! Neither of these points seem to matter much in the face of the data, which strongly points to having your customers stay put in a single line.

So, the next time you’re waiting in line, and the cashier asks you to split into smaller lines, feel free to hold your head up high and ask what’s in it for you. You certainly aren’t doing yourself any favors by complying.

March 6, 2009 | No Comments »

Square Root Day?

I just noticed this article on the front page of Yahoo, which discusses the pending celebrations for tomorrow, in honor of the square root function. Tomorrow is given the name “Square Root Day,” naturally enough, because the date is 3/3/09. Seeing as how there are only 9 square root days per century, apparently the sparsity of this phenomenon is enough to make some people excited when such dates do occur.

Tonight we’re gonna party like it’s 3,996,0011/2.

I don’t want to sound like a curmudgeon, but I’m not really sure who deemed this story worthy of inclusion on the front page of Yahoo. Similarly, I don’t know what it means when the article says that tomorrow’s “holiday” is met with great enthusiasm by “math buffs.” The article seems to suggest that a celebration like this falls within the realms of mathematics, when it more appropriately falls into the realm of numerology.

One could just as easily put significance on equally unimportant sets of dates. For example, I could decide to celebrate the “Doubling Dates” – dates in which the day is twice the month, and the year is twice the day. 02/04/08 has already passed, but it’s certainly not too late to start planning celebrations for 03/06/12, or 04/08/16.

Or, why don’t I start celebrating “Fibonacci Days,” days in which the year must be equal to the sum of the day and the month (those familiar with the process of generating the Fibonacci sequence will understand this choice of name). These days may seem quite plentiful – there will be 8 such days this year alone, for instance – but in any given century, there will be 57 years in a row with no Fibonacci Days! That must make them special, right?

The problem with these sorts of faux “math holidays” is that while they may seem to be an invitation to learn about and explore mathematics, for the general populace they simply perpetuate the stereotype that the bulk of a mathematician’s career is spent multiplying really big numbers together, or trying to find the 101010th digit of pi. This shortchanges the beauty of mathematics and the work of those who make a career out of it.

I’m all for mathematics appreciation days, but if we’re going to have them, let’s have them appreciate something substantive. Why not have appreciation days on the birthdays of some of our most famous mathematical historical figures? Rather than learning that sometimes, the square root of the last two digits of a given year is equal to both the month and day of a given date, which is both uninteresting and limited in scope, why not spend April 15th learning about the contributions of Euler, set aside April 30th to celebrate the work of Gauss, or take September 17th to gain some insight into the prolific work of Riemann? Those with less lofty ambitions could take a day to warn of the pitfalls of combining mathematics with sport by observing the tragic deaths of Galois or Gram, who died in a duel and after getting hit by a bicycle, respectively.

There are enough mathematicians in the history to supply content for every school day of the year, but most students would be hard pressed to name even a handful of important historical mathematical figures. Why this is deemed less important than the observation that 3 * 3 = 9 is slightly beyond me. But then again, maybe I’m just a curmudgeon.

Happy Square Root day, for whatever it’s worth.

March 2, 2009 | 1 Comment »