Math made the headlines last Thursday, with an article about a recent study in the journal Science, which discredits the perceived Gender Gap in mathematics. The AP article can be found here – if you can’t bring yourself to read the article, you can also watch the following clip from NBC Nightly News on the same topic.


The AP article offers a more thorough discussion of the study, which examined standardized test scores for more than 7 million American students. Given the breadth of the study, one hopes it will help dispel any lingering notion girls may have that they are some how innately unable to measure up to boys in math. We do, however, have a ways to go before math professor Barbie starts flying off the shelves.

Any news that can help persuade women to enter mathematically demanding fields is good news. Not only because America needs to retain all the talent it can, but also because every math department party tends to be a huge stag fest. Single ladies, if you’re ever looking for a man, look no further than the event calendar for your university’s local math department. But watch out, if things keep trending the way this article is suggesting, that well will dry up in no time.

One important point that the article makes is that the standardized tests from which the data were collected did not seem to adequately test complex problem solving. This is natural, considering how poorly we are educating students in mathematics. It may be easy, therefore, to dismiss the results of this study with an argument that girls haven’t actually caught up. Instead, the tests have merely gotten easier.

To refute this, I humbly present the following article, regarding a similar study, also published in the journal Science, but with decidedly less fanfare.

Let me highlight the main points of the article:

In search of bridges across the math gender gap, Sapienza and her colleagues analyzed data from more than 276,000 children in 40 countries … Sapienza’s team found that, in more gender equal societies, the gender gap in math disappears. For example, the math gender gap almost disappeared in Sweden (GGI = 0.81), while girls scored 23 points below boys in math in Turkey (GGI = 0.59). Not only did average girls’ scores improve as equality improved, but the number of girls reaching the highest levels of performance also increased…

The research also found a striking gender gap in reading skills. In every country girls perform better than boys in reading In more gender equal societies, the girls’ advantage in reading over boys increases further. On average, girls have reading scores that are 32.7 points higher than those of boys (6.6 percent higher than the mean average score for boys). In Turkey, this amounts to 25.1 points higher and in Iceland, girls score 61.0 points higher.

Said Sapienza, “Our research indicates that in more gender equal societies, girls will gain an absolute advantage relative to boys.”

Are girls as smart as boys? No. Apparently they are smarter. Of course, any girl will tell you such a conclusion is hardly worthy of publication – they already knew as much.

Kudos to these studies for trying to break down some barriers to entry for mathematics. Let’s hope the boys can keep up.

You knew it had to be coming. Any self-respecting individual interested in the intersection of math with popular culture must, at some point, discuss the canonical element of said intersection: CBS’s own crime solving math show, Numb3rs. The use of the 3 is to eliminate any ambiguity surrounding the subject matter of a show called “Numbers.”

Since premiering in January of 2005, Numb3rs has been a consistent performer for CBS, in spite of (or because of, depending on your assumptions about the makeup of the show’s audience) its Friday night time slot. For those of you who may have never seen the show, the following synopsis should help give you some perspective:

Body counts, multiple criminal masterminds, and perpetrators who are likely to act again … this is the world of NUMB3RS. FBI agent Don Eppes (Rob Morrow) couldn’t be more different from his younger brother, Charlie (David Krumholtz), a brilliant math professor at a California university. Don deals in hard facts and evidence, whereas Charlie thrives in a world of mathematical probability and equations. But despite their disparate lives and career paths, Don and Charlie often combine their areas of expertise to solve a wide range of challenging crimes in Los Angeles. (Courtesy of the Numb3rs Season 2 DVD Box)

Indeed, mathematicians would be nowhere without their probabilities, or their equations. Startlingly, one could replace the word “equations” by the word “witchcraft” without at all effecting the tone of the above synopsis.

Let’s take a look at the man who provides the center for the show, Professor Charlie Eppes.

When you’re this good at math, you get
to wear blazers made entirely of gold.
Watch a few episodes of Numb3rs (or read this Wikipedia entry), and you will likely learn the following about this darling mathematician:

1. Charlie graduated from Princeton when he was 16, and is a young math prodigy.
2. Charlie enjoys chess.
3. Charlie
loves blackboards.
4. Charlie has a beautiful girlfriend (a former graduate student, no less) named Amita Ramanujan, who may or may not be related to this famous (and awesome) mathematician.

5. Charlie enjoys socializing, and appears to shower regularly.
6. Charlie loves to explain
mathematical concepts using real world examples, such as vacuums or spiders (see below).

Spiders love math.

What are we to make of these observations? While some play in to stereotypes of mathematicians, others fly in the face of those very same stereotypes.

For the record, let it be known that you don’t have to graduate from college when you’re 16 to be a good mathematician (although it certainly doesn’t hurt). Moreover, not all mathematicians are good at, or even enjoy chess.

By and large, though, we do enjoy a good blackboard.

Points 4 and 5 signify a departure from the math nerd we all know and love. A sexy, brilliant mathematician with an equally sexy, brilliant mathematician? Neither of them even wear glasses! Not to mention the fact that Charlie has more charm than most of the other characters on the show. Is he a mathematician, or a rock star? Or, even better, is he merely a prophet for the future, in which mathematicians and rock stars will be one and the same?

Of course, when you are centering your show on a mathematician, you had better make that mathematician marketable. So in a sense, avoiding certain stereotypes becomes a necessity. Still, having a positive mathematics role model like Charlie Eppes certainly can’t be bad for the math community. In fact, if the below video is any indication, people LOVE mathematicians.

Turn up the volume and press play.
You won’t be disappointed.

This is not to say that Charlie is the only character who displays certain stereotypical idiosyncrasies. In fact, Charlie’s friend and colleague Larry Fleinhardt exhibits behaviors stereotypical of mathematics savants, including a certain social awkwardness, as well as an aversion to any food that is not white. However, since Professor Fleinhardt is technically a Physicist, I will throw him to the Physics camp for interpretation.

Mathematicians may be portrayed relatively favorably, but what about the math itself? Is it legit? Well, it’s hard to say, really, since when all is said and done, there’s not a whole lot of math on display. Certainly there are a lot of scenes with people waving their hands and discussing math, or scenes with chalkboards that have math on them, but these scenes are often placeholders in between scenes with guns or explosions or good looking government employees (here I use “or” in the inclusive, mathematical sense of the word). This, however, is expected, again because of the mass market nature of the program. Overall, I think you’d find it difficult to do math any better after having watched an episode of Numb3rs. On the plus side, they do emphasize that the story lines are based on actual cases, so viewers can take comfort in the fact that even out in the real world, math is helping to bring in the bad guys.

The show certainly doesn’t do any damage to math’s reputation. By making a protagonist who is smart and has a winning smile, the creators seem to be doing their part to show that math needn’t be as scary as it’s made out to be. While there are plenty of moments where it can be hard to suspend your disbelief (he may be smart, but I don’t think even Doogie Howser can solve the Riemann Hypothesis, contrary to what Prime Suspect would have us believe), as an overall ambassador to the universe of mathematics, the show gets a pass. If nothing else, it teaches America that not all mathematicians are completely socially inept, even if we do live at home and put the moves on a few of our advisees.

A well designed t-shirt has the power to delight and inspire; it can break the ice at the start of the evening, and seal the deal at the end. It can be a powerful tool for social interaction, and can help forge the bonds that will last a lifetime.

It is with this in mind that I bemoan the present state of mathematically themed t-shirts, many of which are asinine to the point of nausea. I also feel the need to speak out and warn those who would consider buying such t-shirts for friends or loved ones, under the misguided impression that anyone who studies math will appreciate (much less wear) a t-shirt just because it is related to math.

Let’s analyze some examples, to see just what’s gone wrong with the current state of mathematically inspired fashion. The following five designs can be found here, along with a multitude of others. However, every design seems to fall into one of the categories below:

1) The Symbol Design (epitomized by the pi design):

Many t-shirt designs seem to be of the philosophy that by throwing a bunch of math symbols on a white background, somehow their shirt is clever or interesting. In particular, there are a plethora of t-shirts focused on pi, its decimal approximation, or some pun involving pumpkins or actual pies. Pi seems to have become the poster child for mathematics, which I guess is ok, but I think you will be hard pressed to find a single mathematician who will find the transcription of a long decimal expansion of pi interesting. And for those mathematicians who already feel isolated from society, I don’t think wearing a t-shirt with a whole bunch of numbers on it will make them feel better.

I wouldn’t have as much resentment towards this particular letter if it weren’t for the fact that it always seems to be hogging the spotlight. How about a little love for e? Is it too much to ask for some love on behalf of the Euler constant? Although such similarly themed shirts would be just as unappealing, at least we would have some more variety. Perhaps it has something to do with the fact that other constants don’t lend themselves as easily to puns involving baked goods.

2) The “Why Is this on a T-shirt?” Design:

Let’s be honest: nobody is going to be impressed if you have a t-shirt with a whole lot of formulas on it. In fact, some people might be downright scared of such a shirt. And in a case like the one pictured here, why would one want a list of algebra mistakes on a t-shirt? This hardly seems like the most effective place to keep a list of common mistakes. Wouldn’t a notebook be more preferable? Not to mention the fact that while wearing the t-shirt, it would be difficult for one to read these examples; the fact that there seems to be no attempt at formatting them only compounds the matter. Such examples often make poor use of the t-shirt medium, and would seem to fit much better elsewhere.

3) The Plain Old Not Funny Design:

Come on now. Is this really the best we can do? Not only does this design juxtapose two horribly uninspired slogans, but the pun is terrible. If we’re going to go that route, shouldn’t the shirt say “Math nerd rocks rock!!”? And how can a rock hold a pencil behind its head with no ears?

4) The Minimalist Design:

Sometimes these designs can rise above mediocrity, but I don’t see how they could inspire one to make a purchase. I do appreciate the simplicity of the design pictured here – in fact, of the five displayed here, I would consider this particular design my favorite. It does not smack of effort as do so many other designs, but at the same time, it is this very same feature that keeps this type of design from rising above its brethren.

I would, however, highly recommend the thong version of this item, if you are looking for a gift for that someone special.

5) The Math Crutch Design:

Designing a t-shirt based on mathematics can be difficult: on the one hand, you want to make a joke that people who study math will find amusing, but the shirt should also be decipherable to people who do not do math. Otherwise, why should anyone wear such a shirt in public? Presumably, one wears funny t-shirts to project the impression to others that one has a keen sense of humor. But if many people don’t understand your shirt, what then?

This shirt should not be undecipherable to anyone who remembers some basic calculus – unfortunately, asking that of our citizenry seems to be a bit much. On the other hand, one can be thankful that the calculation is not difficult, so that even if your calculus is rusty you should be able to dust off the cobwebs and compute the joke. However, even though this t-shirt does give one that light bulb moment when the content is understood, the punchline itself is a bit crass for general consumption. Certainly, one should hesitate before wearing this shirt to try and get a rise out of a room full of mathematicians, at least if the mathematicians are senior to the wearer.

* * *

What then, is a fashionable young mathematician to do? Is there no safe haven from uninspired, math-themed t-shirt designs? I think one will find, upon scouring the internet, that resources are indeed limited. However, there are a few rays of hope that shine brightly through the darkness. Perhaps my favorite example is given by the image below.

This design, from threadless.com, is one I would proudly sport on my own masculine frame. It works because it is not actually a joke about mathematics, but a joke taken from pop culture that uses math to deliver the punchline. As such, it becomes accessible to a wide audience, and shows how even rappers can benefit from mathematics. While strong arguments could be made against the plausibility of a linear model as the best fit for the relationship between money and problems, the basic idea is clear, and can be easily understood and admired by people from all walks of life.

Of course, the life of a mathematician can be fraught with disappointments, and this case is no different: the design pictured here has been sold out for such a long time, one wonders if it will ever be restocked. At the same time, one can take solace in the knowledge that somewhere, somebody is looking at this shirt and having a good laugh with mathematics.

Many students often ask their teachers, “Why do I have to learn this boring mathematics? Nobody uses mathematics anyhow.” This new feature, entitled Math Gets Around, will attempt to show you that in fact, mathematics will pop up even in the least likely of places. So learn those multiplication tables, chief.

Today, we see how mathematics has weaseled its way into an unlikely place: the realm of politics. This is particularly relevant given the fact that, as some of you may have heard, there is a presidential election in just a few short months.

Among the general population, there will always be dissidents who complain of the failings of our democratic process. Among these dissidents, you may even find those who question the existence of our two party system, and claim that a system with a larger number of parties would be better for everyone involved. But I am here to tell you the shocking truth: from a mathematical standpoint, this is not the case.

Let me explain what I mean in plain terms. Elections are a part of our democracy. In order to ensure that elections are fair, you would like your process to have certain properties. In particular, any reasonable person should agree that any voting system should satisfy the following three properties:

1) the system should not be a dictatorship – in other words, one person’s preferences can’t be imposed on the results of the election. One can call this the dictatorial property.

2) the system should allow for an individual to rank the candidates in any order imaginable; in particular, any candidate on the ballot should be able to win. One can call this the exhaustion property.

3) the system should be non-manipulable, by which I mean that there are no conditions under which a voter could vote in a manner that does not reflect his or her true preferences in order to achieve the long-term goal of having his or her true preferred candidate win the election. One can call this the manipulability property.

Unfortunately, The Gibbard-Satterthwaite theorem tells us that in any voting system with three or more candidates, and at least two voters, no such voting system exists. In other words, any voting system with more than two candidates must either be dictatorial, non-exhaustive, or manipulable.

Since any system which doesn’t satisfy the first two conditions is impractical, the theorem usually amounts to saying that any voting system you will encounter in real life with more than three candidates must be manipulable.

Oh, but come now, Matt, you might say. Manipulable elections? What hogwash! For this, I turn your attention towards none other than the 2003 Governor election in our fine state of California. Here I quote from the Wikipedia entry on “tactical voting:”

One high-profile example of tactical voting was the situation that led to the 2003 California recall. During the primaries, Republicans Richard Riordan (former mayor of Los Angeles) and Bill Simon (a self-financed businessman) were vying for a chance to compete against the unpopular Governor of California, Gray Davis. As California holds open primaries in which anyone can vote for any candidate he or she pleases, Davis supporters were rumored to have voted for Simon because Riordan was perceived as a greater threat to Davis; this combined with a negative advertising campaign by Davis describing Riordan as a “big-city liberal”, and Simon ultimately won the primary despite a last-minute business scandal. However, he lost the election against Davis; discontent soon led to the recall.

Further examples can be found across the globe (click the link above to read in more detail).


Senators Obama and McCain discuss ways to try and outfox mathematics.

Fine, you might say. But what if we don’t want our elections to necessarily pick winners and losers? Elections, at the end of the day, are merely collections of lists of individual preferences. Is there a way that we can use this large pool of individual data to come up with a preference list that works for the entire community, subject of course to some reasonable assumptions? This subject is taken up in Arrow’s Theorem. The assumptions for the voting system under this theorem are as follows:

1) The voting system should not be dictatorial (see above).

2) The aggregate preference list compiled from individual voting preferences should account for everyone’s vote in providing a ranking for the group, and it should do so in a well defined way – in other words, if two collections of preferences are equivalent (say if person A and person B simply swap their voting sheets), then the ranking for the group should be unchanged. This is referred to as the universality property.

3) Say you prefer candidate A to candidate B, and suppose now that candidate C decides to enter the race. You must alter your preferences to reflect this fact; in other words, how to you feel about C relative to A and B? Whatever your feelings are, when C enters the race, it’s natural to impose the restriction that your preferences for A and B can’t change – for example, if A > B, when C enters the race your list of preferences could be A > C > B, C > A > B, or A > B > C, but not C > B > A, because if you prefer B to A when C is in the race, why wouldn’t you prefer B to A when C is ignored? This property, that preferences for a subset of the candidate list should not contradict preferences for the whole list, is called the independence of irrelevant alternatives, or IIA for short.

4) If everybody in the group prefers A to B, then the ranking for the group should also prefer A to B. This is called unanimity, or Pareto efficiency.

Arrow’s theorem tells us that no such ranking system can satisfy all of the properties given above. Sadly, it would seem that from a mathematical standpoint, no voting system can get it quite right.

However, as with most things in life, there is a silver lining. If you feel our system of elections is broken, don’t worry – you can take solace in the fact that any other voting system you can imagine is probably broken too.

Mathematicians are a rare specimen to behold. While not quite endangered, they tend to congregate in areas less prone to large population densities, such as libraries, or the basements of math buildings, thus making their numbers seem lower than they actually are. This type of behavior is fortunate, for it is because of these tendencies to cluster together that breeding mathematicians in captivity has proven extremely successful – much more so than attempting to breed mathematicians in the wild with the general population (although there are successful cases of the latter phenomenon as well). The point here is that, unless you are of a certain persuasion, you could find yourself going years, possibly your whole life, without ever meeting a mathematician.

Should you be so fortunate to spot one, make sure not to approach too quickly, or you may scare the mathematician away. If you are vigilant, you may be able to engage the mathematician in conversation. In the event that this happens to you, there are a few common misconceptions about what’s appropriate to say to one who studies math. For your consideration, I present three common things people say when they find they are talking to a mathematician, and why one should avoid saying such things.

1) “You study math? You know, I never liked math.”

Ok. What exactly am I supposed to say to that? If you met a musician, would you say “You know, I’ve never liked music. Not my cup of tea.”? If you met a lawyer, would you say, “Yes, studying the law is all well and good, if you are a sucker, that is.” I should hope not. But when it comes to mathematics, people feel it is entirely acceptable to share with you how much they despise that which you devote so much of your time to. Perhaps they were abused by mathematics early in their life (in fact, for many I think this is the case). However, I am not your psychiatrist, and I have no interest in telling you that it’s ok, the math will not come out and bite you now that you are grown. Remember, when it comes to mathematicians, if you don’t have anything nice to say, tell them how good they look in their glasses.

2) “You study math? What’s 12,147 times 5,382?”

Of course, equally atrocious is any question of the form “What is n times m?” where n and m are sufficiently large. Contrary to the image of mathematicians portrayed in most popular media, we are not all socially awkward computational savants – at the very least, we are not all computational savants (even if we do have to buy all our underwear from the K-Mart on Oak and Burnett). Very little, if any, of what mathematicians do involves finding the product of very large numbers. And should we need to multiply very large numbers, we have calculators, just like the rest of you. If we are able to compute larger products than you, it is usually only because our calculators are more expensive. This brings me to a related point, of particular importance when you are out with a large group of people in a restaurant or bar, and amongst you lies a mathematician:

2.5) “Give the tab to the math guy! He’ll figure it out!”

It is precisely these kinds of prejudices that have kept mathematicians down for hundreds of years. From personal experience, under the pressure of society’s expectations on my computational abilities, I find it more difficult to figure out how to split the tab when everyone is eyeing me, wondering what is taking so long. But let’s be honest: you’re just trying to find an excuse not to figure it out yourself. Because nobody wants to figure it out. Well, mathematicians are a lot like assassins – they will do your dirty work, but they expect to be paid top dollar for their services. So the next time you think to ask a mathematician to tell you how much you owe, be careful. You may be paying a hefty, hidden premium for his or her services.

While I’m at it, we don’t enjoy doing our income taxes any more than the rest of you, either.

3) “So tell me exactly what kind of math you do.”

This one is a bit subtler. In certain contexts, such a request will endear you in the heart of a mathematician for many years. But often times, especially in party settings, such a remark is made simply to try and start a conversation. Gentle reader, before you make such a request, you must be made aware of what a deep rabbit hole you are jumping into.

The work of a mathematician tends to be extremely specialized. Specialized to the point where mathematicians will not understand what one another is doing. So unless you want an answer so general as to make it mostly useless (“Number Theory,” “Analysis,” “Geometry,” etc.), you are, in effect, demanding that the mathematician compress years of difficult study into a brief and easily comprehensible tablet. Now, of course there is something to be said for developing the skill of being able to translate one’s research to a level that can be understood, at least in a broad sense, by a large population. But at the same time, for some things, it is simply not possible to have a simple explanation up one’s sleeve. This is why mathematics is difficult. So an explanation of current research will, more often than not, take time, and be at least slightly incomprehensible.

This is not so much a problem, if one is willing to listen. The problem arises when, as mentioned before, one has merely made this remark to try and start a conversation. Not knowing what the questioners have gotten themselves into, they now must sit and listen to a mathematician talk at length about things the questioners really may have had no interest in to begin with. If this goes on too long, it can create tension, and furthermore, when the listener stops paying attention, the mathematician’s time is essentially being wasted. That time could be better spent doing research, or getting all the digits from more attractive party attendees.

In summary, I know it can be difficult knowing what to say when you meet a mathematician. But there are certain things you should not say, at least without understanding what you’re getting yourself into. It may help to pretend that instead of a mathematician, you are talking to a rock star or international supermodel – this should not be a difficult stretch of the imagination, and will help you avoid the taboos mentioned above.

With the blessing of Gov. Arnold Schwarzenegger, it looks like the great state of Kah-lee-foh-nya is set to require all 8th graders to take Algebra I. The Sacramento Bee has an article regarding this development, see the link here:

http://www.sacbee.com/1089/story/1072095.html

A valiant effort, to be sure. But what good does this do, aside from making for good press? Let’s examine.

Recalling, first of all, that the California public education system is typically ranked somewhere in the lower forties (and recalling also that, given there are but 50 states in the union, this makes California’s ranking fairly dismal), it is natural to ask what the short term impact of this new requirement will be. If we’re looking for a sustainable solution to the education crisis in this state (forgetting for a moment what’s going on at a national level), this hardly seems to fit the bill. There’s no mention in the article about how such a proposal will be funded – a fairly basic question if such a revamp is to occur. How will we pay for training current teachers or hiring new teachers to meet the larger demand for Algebra I instructors? How will we get this current crop of students up to snuff, so that when they reach 8th grade and are thrown into an Algebra class, they are not completely lost? Now, don’t mistake this as a statement that teaching Algebra to 8th graders is a bad idea – far from it. But saying 8th graders should unequivocally take Algebra I seems a bit like saying developers should build houses on sand.

Mathematics is a tiered discipline, one that relies heavily on earlier results. With a mastery of multiplication, a delight with division, a familiarity with fractions, and whatever other mathematical alliterations one can muster, algebra itself is not such a difficult thing. However, all it takes is one substandard teacher, or lackluster book, or even an illness that takes a student out of the class for an important lesson, and suddenly that math train derails, sometimes leading to injuries that can be life-lasting.

What to do? Of course, the easy answer is to put your money where your mouth is. If we want students to understand Algebra in 8th grade (this is very different from wanting them to take Algebra in 8th grade), then invest in the infrastructure to do so. Pandering to some standards committee with something like this is not only a bit insulting, but also does yet another disservice to the students in our public education system – and really, haven’t we done enough already?

On a related note, the tone of this article in particular does little to buck the stereotype that mathematics must necessarily be a sisyphian task. My favorite line is this one, in regards to the proportion of 8th graders who currently take Algebra I: “About half the state’s eighth-graders now complete the tough subject.”

Not the subject, mind you, but the tough subject. God bless those 8th graders willing to accept the arduous challenge of Algebra I. Never mind the fact that students of comparable age from all around the world are murdering us when it comes to math ability. Algebra must be tough, because most of us can’t do it.

How about just calling it a subject? Do we really need to demonize Algebra further? Hasn’t she endured enough? Calculus is bigger, and can take the punches. But Algebra? She’s just a puppy dog. Our system of math education is just too broken for many to notice it.