Restructuring the Math Pyramid?

A friend recently shared with me the following video from TED (see below). In it, mathematician (or, in this case, mathemagician) Arthur Benjamin gives a brief argument for eliminating calculus as the top of the “mathematical pyramid” in high school education, and replacing it probability and statistics. The main reason for this shift is that unless you are planning to have a career in a technical field, it’s unlikely you’ll find a use for calculus in your everyday life, but an understanding of statistics can benefit you no matter what you do. For example, it can help you to build an intuition about day to day decision making when risk and uncertainty are involved. Here’s the video (it’s short, only a couple of minutes):


A noble goal, to be sure, and it’s certainly a solution that wouldn’t cost a whole lot. There is an argument to be made for such a change lurking in here somewhere, but coming in at under 3 minutes, Benjamin’s argument barely scratches the surface. In no particular order, here are some of the problems I have with his proposal:
1) Arguing that students shouldn’t learn calculus because they may not use it in their everyday life is specious. By this reasoning, I should never have taken any courses in history, biology, or chemistry. The purpose of high school education in this country seems to be not only determining what educational avenues students want to pursue further, but also what avenues they don’t want to pursue. If you want to argue that students should only be learning things that they can apply to their everyday lives, then you are arguing for a much more sweeping reform of education.

I do acknowledge that there is an opportunity cost at work when we spend a year teaching a student calculus rather than statistics, and certainly the average student will find more use later in life for the latter. But there’s also an opportunity cost at work when we spend a year teaching a student statistics rather than calculus, especially for students who aren’t sure in what direction their academic future will head. If anything, this seems to be an argument for offering both statistics and calculus for students, rather than forcing them into one option or the other.

2) About 2/3rds of the way through the talk, Benjamin asserts that “if our students, if our high school students, if all of the American citizens knew about probability and statistics, we wouldn’t be in the economic mess we’re in today.” This is met with some cheers from the audience, but is it actually true?

The answer depends on your definition of “knowing” probability and statistics. I agree that having some knowledge of statistics is a good thing for the population at large, and there are no doubt many fundamental principles that could be taught at a high school level – for example, the idea that correlation does not imply causation, or the ways in which one can manipulate data or graphs of data. These topics, among others, are discussed in the book How to Lie with Statistics, which would be a great required reading book for any teacher trying to impart intuition and a healthy dose of skepticism onto his or her students, and is written for a general audience.

However, even if everyone in America had this basic level of knowledge, it’s not at all clear that this would have somehow saved us from economic catastrophe. If you work in finance, odds are pretty good that you already have a knowledge of statistics that goes beyond a high school level, but this didn’t stop the economy from tanking.

Nassim Nicholas Taleb wrote an excellent article last year on the limits of statistics, which is well worth a read if you can spare the time. One of the arguments he makes is that part of the reason the financial models caused such an economic implosion was that these models are necessarily unable to predict black swan events, which can have a tremendously negative impact, but are also tremendously rare. In fact, he argues that statistics is actually quite poor at trying to predict what will happen in extremely complex systems where rare extreme events can have a profound effect on the system.

However, this is not what one learns in a high school or even undergraduate class on statistics. Most problems at this level involve simple systems (games of chance, for example). In other words, studying statistics at a low level does not expose one to the subtleties and limitations of the subject – in particular, I don’t think it’s feasible to say that if every high school graduate had taken a course in statistics, somehow we would have prevented the current economic catastrpohe. To do so would have required a much deeper understanding of statistics among those applying the financial models than can be supplied at the high school level.

This brings me to my third point…

3) To have a good understanding of statistics, one must already have a working knowledge of calculus. There is a limit to the amount of depth a probability or statistics course can explore when calculus is not a prerequisite, and because of this many results (such as the Central Limit Theorem) are stated without proof. This is fine if you are simply trying to expose students to some of the standard tools in the subject, but if you can’t go deeper, there really is a limit to the level of understanding a student can achieve.

I agree that there is a great deal of value in teaching statistics to high school students, even at the level of pre-calculus. One can still impart a significant amount of intuition at this level. However, for students who plan to use statistics in any significant capacity, it’s important that they develop a working knowledge of calculus as well.

If you’re not planning on going into a technical field, certainly you’ll get more value out of a basic statistics class than you will a calculus class. But students who dislike math in general will probably still dislike statistics, even though there’s more to like for someone who’s not interested in math than there is in a calculus course.

This, in turn, brings me to my next point…

4) In the larger debate over the failings of math education in America, the choice of whether to teach statistics or calculus in secondary school misses the point entirely. By the time students reach the later years of their high school career, most already have a pretty well developed sense of their relationship to mathematics – either they had good teachers and enjoyed the subject, or through a series of misfortunes which may have been out of the student’s control, they feel like math is a subject they will never understand, and will struggle with until they have the freedom to not take a math class, and are finally free from its iron grip.

Sadly, the problems with math education in this country run much deeper, and swapping out calculus for stats at higher levels won’t alleviate the fundamental problems students have with mathematics. When I grade papers in a calculus class, students make just as many (if not more) algebra mistakes as they do calculus mistakes. In other words, many students leave the year without having mastered the math concepts presented to them during that year. Compound this over several years, and it doesn’t matter if you give them a calculus book or a statistics book – they will have trouble because they haven’t mastered the prerequisites.

Certainly one can argue that there are fewer prerequisites in a statistics class, but prerequisites are still present, and algebra is certainly one of them. If a student has a poor understanding of algebra, it’s reasonable to assume he will have significant gaps in his understanding of statistics, and if the goal is to give students an intuition for randomness and understanding data that can help them in their everyday lives, gaps in statistical understanding are significant problems. Therefore, achieving this goal isn’t as simple as making sure every high school senior has taken a statistics class – we really need to insist that every student first has a working knowledge of algebra. This is a problem we have already, and is not resolved by Benjamin’s proposal.

5) This is a small point, but important. What really bothers me about this talk is when Benjamin makes the statement that, “If [probability and statistics] is taught properly, it can be a lot of fun!” Well, yes, but this is true of any subject. Implicit here seems to be the idea that calculus cannot be fun, even if taught well. I’m sure this isn’t Benjamin’s intention, but it’s easy to misinterpret, especially if you are someone who has never taken a calculus class, or has fallen victim to the commonly held opinion that calculus is some kind of black magic whose secrets only a chosen few can hope to unravel.

The truth (and one that Benjamin knows) is that any math class can be fun if taught properly. A more accurate statement might be “it’s easier to make probability and statistics fun for students,” because of the vast applicability to everyday life, from games of chance to calculating the probability that someone in a family is colorblind. But to suggest that statistics is inherently more fun for students than calculus does a disservice to all the great teachers of calculus. Either class can be fun and valuable if taught well, or traumatizing if taught poorly.


Mr. Prezbo, using probability to make math fun. No doubt he could work this magic on other math subjects as well.

I understand what Benjamin is saying, and I also understand its appeal. The argument works well as a 3 minute sound clip, but upon further reflection, there are some significant questions that need to be addressed. There are many problems with math education in this country, and I’m not sure which, if any, are solved by this proposal.

From my own experience, no students in my high school were forced to take either calculus or statistics, although both courses were offered. Preparing students exclusively for either one or the other will of course do a disservice to some, so perhaps putting both on the table is the best compromise, although this becomes a problem for schools with limited resources. I am confident, however, that simply putting statistics on the pedestal currently occupied by calculus doesn’t do a whole lot in terms of fixing everything that’s broken.

July 22, 2009 | No Comments »

Math in the Movies: Revolutionary Road

This past week I watched Revolutionary Road, the Oscar nominated 2008 film directed by Sam Mendes. The film stars Leonardo DiCaprio and Kate Winslet as a highly dysfunctional couple named the Wheelers, who live in 1950s suburban Connecticut. For those of you who may not have seen this feel-good picture, here’s a trailer:




The trailer doesn’t address the question of what this film has to do with mathematics. The answer lies in the character of John Giving, a “mathematician” played in the film by Michael Shannon (who turned in an Oscar-nominated performance).

We first hear of John Giving from his mother, who informs Mrs. Wheeler that her son has a brilliant mind, as evidenced by his PhD in mathematics, but that he has been institutionalized, and his doctors have suggested that it would be good for him to go out and make some friends. This introduction did not bode well for the film’s representation of the mathematically inclined, but how did the rest turn out?

Let’s explore the usual stereotypes.

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

This is probably the most common stereotype in films about mathematicians, and this film would certainly have us initially believe that it is no different. To be fair, we’re never told that Dr. Givings was ever an especially gifted mathematician, but the first facts we learn about him is that he is a mathematician, and he is insane.

But is he really? He’s certainly outspoken, and doesn’t fit in with the established conformity that has become synonymous with 1950s American life, but we’re never given any clear indication as to whether he truly did or did not belong in an asylum. Indeed, the film seems to play on the convention of the insane mathematician – Dr. Givings is told by society that he is insane, but his outspoken attitude sometimes makes him seem like the only sane one in the film.

In fact, it’s not at all clear that he is even a mathematician – perhaps he has just been given this label by a mother who yearns to take pride in her son. When Frank Wheeler asks Givings about his background as a mathematician in the book on which the film was based, Givings asserts that he is not a mathematician. “Taught it for awhile, that’s all.”

I’ve decided to give the film a pass on this stereotype. Givings may be insane, but he certainly seems more lucid than anyone else living in the suburbs. +0.


- People who are good at math are socially awkward.

Givings is awkward, but not in the way one would expect in a portrayal of a mathematician. Instead of being nervous in social situations, he seems to relish them, and is quite outspoken in his opinions. The awkwardness therefore stems from his seeming inability to keep his mouth shut.

He does create awkward situations, but not in the way you’d expect given the stereotypes about mathematicians. So I’ll give this one a pass as well.


One of two scenes featuring John Givings.

Related to the notion of social awkwardness is the idea that people who do well in math or science are not good at picking up on nonverbal cues. However, as the clip above illustrates, Givings is quite good at picking up on these cues – better than anyone else in the film, in fact. That he can observe these cues and still do mathematics is a good thing, although to be fair, Givings says that his mathematical abilities disappeared following his shock therapy treatments. +1.

Givings is certainly not your stereotypical mathematician. I worried that this film would play in to all of the stereotypes surrounding folks who do math, and so I was surprised to find that the film played with these stereotypes in a way that one doesn’t usually see. I prefer this performance to other Oscar nominated crazy mathematician roles, but I still long for the film that shows me a mathematician who’s just a normal dude (or, even better, dudette).

July 17, 2009 | 2 Comments »

Math Gets Around: On Birthdays and Trading Cards

Today marks the 1 year anniversary of Math Goes Pop! I started on somewhat of a whim after reading an article about compulsory Algebra I education for all California 8th graders (although what with our finances down the toilet, who knows what the current status is here). When I started writing I wasn’t sure there was enough content out there to sustain a blog with this one’s focus. Once I started digging, though, I found that the rabbit hole went quite deep, and so here I am a year later with plenty left to talk about (the recent obsession with pointless math holidays certainly has helped with my output).
Given the date, it seems fitting to begin by mentioning the birthday problem. This is a standard problem given in any introductory probability course, but many people find the result counter intuitive at first.

The birthday problem asks a simple question: if you have a room full of people, how many people do you need so that there’s a 50% probability that at least two of them share the same birthday? In the simplest setting, we make a few assumptions: for example, we usually ignore the presence of leap years, and we assume that each day of the year is equally likely to be someone’s birthday.

With these assumptions, solving the problem is not difficult. The probability that at least 2 people in a group of n share a birthday is 1 minus the probability that all n people have different birthdays. In other words, the probability is

1 – (1 – 1/365)(1 – 2/365)…(1 – (n-1)/365).

This is because the second person has a 1 – 1/365 = 364/365 chance of having a birthday different from the first person, the third person has a 363/365 chance of having a birthday different from the first two people, and so on.

So, to answer the question, we just need to find the smallest n such that the expression above is greater than 1/2. The punchline is that n only needs to be 23 or more in order for this inequality to hold. In other words, in any room with at least 23 people, there’s a better than 50% chance that two of them have the same birthday.

For many people, this number seems too low – after all, you may have been in several groups of 23 or more and never or rarely met someone with your birthday. The problem with this argument is that it addresses a different question. The reason why we only need 23 people to have a 50% chance of finding a common birthday is that we are placing no restriction on the date of the shared birthday. Requiring that someone share the same birthday as you fixes the date, and this changes the problem.

If instead you want to know how many people you need in a room so that there’s a 50% chance one of them will have the same birthday as you, this probability is given by the new expression

1 – (364/365)n.
To make this greater than 1/2, we need to take a much larger n: n = 253, to be precise.

As I said, this is a well-known problem in probability theory. A few months ago, however, I was asked about a variant of the birthday problem by my friend Gabe over at Motivated Grammar. Rather than looking for identical birthdays, what if you look for different birthdays? More specifically, how many people do you need in a room to guarantee with, say, a 90% certainty, that every day of the year is someone’s birthday?

Of course, you could always pick poorly so that everyone in the room has the same birthday, but the odds of this happening are quite low. In fact, this problem is more commonly known by another name: the coupon collector’s problem.

For this problem, suppose you are clipping coupons from a newspaper (any newspaper except for USA Today). There are n different coupons you can collect, but each newspaper only has 1 coupon, and you can’t see which coupon the newspaper has until you’ve bought the newspaper. In this context, the question becomes: what’s the probability that you’ll need to buy at least newspapers to collect n coupons?

If you prefer, you can think about this problem in terms of trading cards as well. Each pack of cards is akin to buying a set of newspapers, and you want to know the probability that you’ll need to buy at least a certain number of packs in order to collect all the cards. From baseball players to Pokemon, this same problem governs the distribution (assuming that all cards in the back are equally likely to be in the pack, which may not always be the case).

What’s known about this problem? Well, as I said above, even if you buy hundreds of thousands of cards, or stuff millions of people in a room, there’s no guarantee that you’ll collect every card or every date. However, on average, the number of cards you’ll need to go through to complete a set of size n is about n*logn.

In terms of birthdays, this says that if you want to collect every date, on average you’ll need to pool together around 2,153 people. Why such a large number? It’s not unreasonable to expect something like this – when you first begin collecting people, it won’t be hard to get people with different birthdays. However, as your numbers increase, you’ll get a new birthday less and less frequently. Finding that last birthday could prove to be quite elusive.

The same analysis works for trading cards. Trying to complete your collection of series 2 Teenage Mutant Ninja Turtles Animated Series trading cards? Well, my friend, with 88 cards total and 5 cards per pack, you can expect to buy around 79 packs of cards. Perhaps this box set would be a better investment.

Mondo to the max, indeed.

While the expected values are easy to calculate, it may be that you need to greatly exceed the expected value in order to complete your collection. However, one can use Markov inequalities to get bounds on the probabilities. For example, there’s a 90% chance that you’ll be able to hit every birthday if you take no more than about 21,535 people. To bump those odds up to 95%, take 43,069 people.

So, for parents whose children who have gotta catch ‘em all, you can use these methods to get a rough estimate for how much that completion will cost you. And if you’re trying to get a room full of people together so that every day of the year is someone’s birthday, I’d strongly suggest not picking people at random. What an awkward party that would be.

July 9, 2009 | No Comments »

USA Today, you are also on my list

It bothered me when USA Today, in an article celebrating “math holidays” centered on the numerology of certain dates, linked to a post I had written about how these holidays are stupid, without even mentioning my contrary opinion. However, I was willing to let it slide, since I was able to say that I was linked in an article from USA Today.
However, an article posted today is just too much. USA Today, you have officially made it onto my list.

The headline for the article really speaks for itself: “Rare time/date alignment could mean opportunities.” This refers to the fact that in the wee hours of the morning today, it was 4:05:06 on the date 07/08/09.

Money quote:

Although the alignment may not mean anything specific, it could be a good day to do something for yourself and others, said Betsy Carlson, a Palm Springs tarot card reader and numerology expert.

“It’s a good day to make money and have good health,” she said.

When is it not a good day to have good health? Who wakes up, looks out the window, and decides that no, today is really a day for rather poor health? What does this even mean?

And how can someone be a “numerology expert?” Would any self-respecting newspaper publish a story from a phrenology expert? Why does numerology so often seem to get a pass? If you want to know why nobody reads newspapers anymore, this serves as an excellent indication. Is this seriously what passes for journalism in 2009? I guess because it’s in the “Offbeat” section, that makes it all ok.

Money-er quote:

Joy Meredith, owner of Crystal Fantasy in Palm Springs, Calif., noticed the alignment, but she’s more focused on this morning’s lunar eclipse, she said.

Nonetheless, she’s a fan of numerology and sometimes tries to determine if numbers have meaning.

“I feel they could be significant, so I’m looking for that,” she said. “If they’re not, they’re not. But I am looking to see if there is any significance.”

Dear Joy, I think I can help you out. I have discovered that numbers do indeed have meaning. The meaning of the number 12, for example, is the number of eggs you’ll get if you go to the store and buy a dozen eggs. The number 1 represents the number of newspapers that thought such a crackpot story was worth publishing. And so on.

Here’s another question: how does one go about “looking” for significance in a given “time/date alignment?” What oracle does one consult in search of insights into the mysterious nature of 4:05:06 07/08/09? God, I hope USA Today follows up on this article, so that all of my burning questions can be answered.

The worst thing about this article is that if you’re going to post garbage, at least post the most interesting garbage possible. The fact that 4:05:06 07/08/09 occurs today (or slightly less than a month from today, for those of us outside of the states) is not nearly as interesting as the fact that 12:34:56 07/08/09 occurs (twice!) today. If you’re into this sort of thing, I see no reason to find the first time/date alignment more interesting than the second. I must admit, even I cannot resist publishing this post at the appropriate time.

Number partyyy!!! Courtesy of Kitsune Noir, by way of Meebobebo.

Watch out, USA Today, because I’ve got you in my sights. My influence is vast, and my resources infinite. Let’s dance.

July 8, 2009 | 1 Comment »

Ron Gordon, you are on my list

Not this again. I’ve now discovered that the mastermind behind these so-called math “holidays” is a teacher named Ron Gordon. Not only was he the one to spearhead the Odd Day initiative 2 months ago, but he’s gone so far as to double dip and call today Odd Day as well, citing the fact that standard date notation for most of the world is DD/MM/YY, rather than MM/DD/YY.

Thanks for double dipping, Mr. Gordon, so that I can read these pointless articles yet again. Mr. Gordon has even set up a web page and a contest, with cash prizes for those who can celebrate Odd Day the most enthusiastically. Needless to say, I don’t think I will be the recipient of any such prize.

The road to hell is paved with good intentions, Mr. Gordon. I’m just sayin’.

July 5, 2009 | 2 Comments »

Decimal Point Fail, Ctd

I apologize for my silence over the past few weeks – I have been out of the country learning math and eating pancakes. While I get back into the swing of things, I’ve got a couple of points to mention that relate to earlier posts regarding our collective inability to correctly use the decimal point.

The first is a picture from a flyer advertising maid service. Here’s the ad (sent in to me by a dedicated foot soldier in the army that is my readership, a.k.a. my mother):

Names and phone numbers have been cropped out to protect the innocent. But in a case such as this, are there really any innocents?
Although we’ve seen decimal point errors on signs before, this one is arguably the most egregious of all. Presumably the intended price is $100 – if that’s the case, then not only is the decimal point in the wrong place, it’s not even necessary. It’s hard to imagine how this mistake could’ve been made and then gone unchecked, but if you live in San Francisco and are looking for some cheap maid service, I can definitely hook you up. Also, if anyone else has pictures which evidence a lacking in mathematical proficiency, feel free to send them my way.

On a related note, while we all knew that Verizon employees suck at math, apparently this low tolerance for mathematical ability among cell phone providers spreads even wider. More specifically, there is evidence that AT&T employees suck at math, too.
This fact has been brought to us courtesy of MythBusters co-host Adam Savage. According to this article, at the end of last month Mr. Savage was charged $11,000 for a few hours of web browsing while in Canada. This figure alone should be enough to make us skeptical of the math at work, but what’s worse is that when customer service tried to explain the charges, they told savage that “data is charged at .015 cents, or a penny and a half, per kb.”
Depicted here is the effortless charm and confident sophistication that comes with a knowledge of mathematics.

Sigh. Perhaps it’s time to switch to T-Mobile?
July 2, 2009 | 2 Comments »