Update (October 2009): I’ve written a follow-up article with more math themed costume ideas.
With Halloween but a few short days away, many of you with a love for both dress-up and mathematics are probably thinking hard about what you should be this year. I thought it would be fun to find some good math inspired Halloween costumes using the transformative power of the internet, but unfortunately there really wasn’t much to get excited about. After spending some time scouring, the only costume ideas that were even tangentially related to math that I could find were the following:
By far the most offensive of these costume choices. Of course, this offers a broader stereotype than that of the mathematician, but the mathematician and the nerd trade at about the same cultural currency value.
What’s most disappointing is the fact that this isn’t even a top quality impersonation of your stereotypical nerd. Anyone thorough enough to make use of the pocket protector would almost certainly keep their pens looking tidier than this guy does.
I guess if you want to reinforce 1980s stereotypes about smart people this is the costume for you. However, I feel like this is right on the border of even being considered a costume. If someone feels so lackadaisical about dressing up, I think it would be more likely for them to simply not do anything.
This one is also fairly uninspired, although it certainly does have simplicity and transparency going for it.
In this case, the real irony is that math teachers do not wear shirts proclaiming that they are math teachers, so anyone who wears this shirt must paradoxically not be dressed in a math teacher costume.
Really, a more convincing math teacher costume would just be a normal, everyday outfit. But in this age of Halloween spectacle, we all know that such a costume simply won’t cut the mustard.
This is the best costume I could find, although its link to mathematics is more tenuous than I would prefer. Taken from Discover’s Top Ten Science Halloween Costumes, the Rubik’s Cube Head certainly could lead to a conversation about mathematics, although I find this unlikely.
What’s more likely is that people will ask you things like, “How can you see out of your giant cube head?” The more rambunctuous may pin you down and try to see if the cube is actually functional. For a select few, however, the discussion may turn towards mathematics of the Rubik’s Cube, and at the very least, you should score some points for originality.
Sadly, in general the top ten science costumes from the blog skew heavily towards the science side, and away from the math side. The only other costume from the list that shares a close connection with mathematics is number 3, the Cat-ahedron, which, though charming, I have omitted since to the best of my knowledge, none of my readers are cats.
It is clear that the costume ideas available on the Internet are limited. For those of you wishing to combine math and Halloween, I wish you the best of luck. Do any of you know of costumes that have succesfully synergized the two?
Earlier this month, the New York Times ran an article about the dearth of U.S. students with strong skills in mathematics. While this is not quite a revelation, it is made more timely by the recent release of a study that looked at data from Putnam exams, International Mathematical Olympiads, and data from other programs meant to nurture younger students in mathematics.
This type of data is more powerful than looking at SAT scores, for instance, because exams administered in a mathematics competition are notoriously difficult. There are thousands of students who will score an 800 on the math section of the SAT, and so this test offers no way to distinguish between them. Looking at this other data, however, allows us to gain a much deeper insight into the abilities of students in the U.S. with an aptitude for mathematics.
The data suggests a couple of things. First, contrary to the Gender Gap theory I have discussed before, there are many women who perform extremely well on these exams. While the data can’t support or refute the Gender Gap theory conclusively, it does show that indeed there do exist women who are good at math.
The second conclusion, which is somewhat broader, is that a majority of students in the US who excel in these exams are either foreign (for example, the Putnam exam can be taken by any undergraduate in the US or Canada, not just citizens), or are children of immigrant parents.
The combination of these two points is highlighted in the breakdown by country of the women who compete in these types of competitions. The article informs us that, regarding the makeup of the teams sent to the International Mathematics Olympiads,
All members of the United States team were boys until 1998, when 16-year-old Melanie Wood, a cheerleader, student newspaper editor and math whiz from a private school in Indianapolis, made the team. She won a silver medal, missing the gold by a single point. Since then, two female high school students, Alison Miller, from upstate New York, and Sherry Gong, whose parents emigrated to the United States from China, have made the United States team (they both won gold).
By comparison, relatively small Bulgaria has sent 21 girls to the competition since 1959 (six since 1988), according to the study, and since 1974 the highly ranked Bulgarian, East German/German and Soviet Union/Russian IMO teams have included 9, 10 and 13 girls respectively.
The data is troubling because not only does it show that Americans are getting trounced on the international stage, but it shows that when we do excel, it’s often because of imported values from the countries that are trouncing us in the first place.
This leads to an important question: Is American culture to blame? Why do our students simply not perform as well?
Indeed, most people interviewed seem to think that culture is, if not the primary cause, certainly a guilty party. Simply put, mathematics is held in a much higher regard in other countries. Consider this explanation of the perception of mathematics in China:
Dr. Feng says that in China math is regarded as an essential skill that everyone should try to develop at some level. Parents in China, he said, view math as parents in the United States do baseball, hockey and soccer.
“Here everybody plays baseball,” Dr. Feng said. “Everybody throws a few balls, regardless of whether you’re good at it, or not. If you don’t play well, it’s O.K. Everybody gives you a few claps. But people don’t treat math that way.”
If we want to tackle this problem, looking for solutions from the cultural side shouldn’t hurt. There are many negative perceptions that keep math out of the cultural consciousness, not least of which is the idea that somehow mathematics is meant to be tedious, difficult to understand, and without application. If other countries can highly value mathematics, see the use for it, and believe that anyone can achieve a certain level of mathematical sophistication with due diligence, surely America can as well.
Of course, getting our culture to that point will require serious work. However, certainly there must be some baby steps that will help us along the way. With that in mind, here are some suggestions that may help bring U.S. culture and mathematics into a more harmonious relationship.
1. Get a mathematician on the Wheaties box.
The analogy between math and sport is certainly a rich one. Both require hard work and discipline in order to excel. Both should be included in any child’s education. And both attract people to their summer camps.
At the same time, there is quite a wide cultural divide that is perceived between these two groups. Athletes are put on pedestals (literal and metaphorical), and their toned physiques are heralded as the pinnacle of human achievement. They are also widely regarded for their dedication, their determination, and nobody questions their hygiene. Sadly, the same cannot be said for mathematicians.
To combat this inequality, why shouldn’t it be the case that top performers from both fields should be able to have their face on a Wheaties box? Certainly breakfast is the most important meal of the day for both athletes and mathletes – we should emphasize this point by highlighting the achievements of mathematicians on the orange box we all know and love.
That’s a nice throwing arm you’ve got there, Josh Beckett. But how’s your multivariable calculus?
2. Put a mathematician on The Simpsons.
Sure, The Simpsons doesn’t hold quite the cultural sway that it used to, but its longevity shows that it has carved out an enduring place for itself in our culture. It is still quite popular, and has its share of devotees, and for this reason many people still pay attention to what it has to say, even if it may have been eclipsed by other series (animated or otherwise) in recent years. In this respect, it is a bit like the Hillary Clinton of prime time television.
Therefore, it stands to reason that having a mathematician lend their voice to an episode of The Simpsons, if handled in the right way, certainly couldn’t hurt to flip the cultural perception of mathematics on its head. The Simpsons has featured hundreds of guest stars: celebrities, heads of state, authors, athletes … the list goes on. Moreover, a move to bring in a mathematician would not be entirely unprecedented – Stephen Hawking has made not one, but two guest appearances on the show, and although he is a physicist rather than a mathematician, it would not be such a huge leap to move from a guest star of the former occupation to a guest star of the latter.
Of course, we could give the show the benefit of the doubt, and assume that despite their best efforts, producers have been unable to find mathematicians who would be willing or able to participate. Should this be the case, I am willing to humbly submit myself for such duties. I believe I am able to shoulder the tremendous responsibility that such an opportunity would entail.
Stephen Hawking: Bringing theoretical physics to the forefront of pop culture since 1999.
3. Endorsements.
How do athletes and celebrities become cultural icons? Certainly their abilities take them far, but would Tiger Woods be as well known without his lucrative contract with Nike? Would Michael Jordan have been as successful without his deals with McDonalds, Coca-Cola, Hanes, and most importantly, Ball Park Franks? Would Gary Coleman be where he is today without Cash Call? I think not.
Given all of this success, there’s no reason why academics shouldn’t be able to dip into the same pot. You just got tenure at a major university? Well congratulations, here’s a contract with Gatorade. Solve the twin prime conjecture? Then you get to sport the 2009 Saturn Astra!
Of course, it’s a slippery slope to begin mixing economic incentives with academic achievements. But it certainly would help propel academics, and mathematicians in particular, into the spotlight.
*
Ok, so I’m (mostly) kidding about the above suggestions. But this disparity between our cultural views of mathematics compared to the views of other countries really is troubling, especially as we look towards a future that demands more and more technical sophistication from its populace. Bringing mathematics out of the cultural doghouse requires more transparency on our part, so that people can see why mathematics is important, and it also requires a better educational foundation, so that students see math as something beautiful and widely applicable, rather than some draconian set of rules, the knowledge of which was rendered obsolete with the creation of the calculator.
With the right resources, we can turn this perception around. Until then, be on the lookout for any chance to defy the stereotype that math isn’t worth knowing. Every little bit helps, and every little bit will be needed.
As a footnote, those of you who read the NYT article will notice that it mentions a particularly hard problem from the 1996 IMO exam. So difficult was this question that only 6 students out of a pool of hundreds were able to completely answer it. If you’re looking for a good way to spend an afternoon, here is the problem in question (if nothing else, you can learn a new word by reading it):
Let ABCDEF be a convex hexagon such that AB is parallel to ED, BC is parallel to FE and CD is parallel to AF.
Let RA, RC and RE denote the circumradii of triangles FAB, BCD and DEF respectively.
Let p denote the perimeter of the hexagon. Prove that
As many of you with Gmail accounts may already know, Google launched a feature last week that aims to put arithmetic squarely in the shoes of your most trusted wingman. The feature, dubbed Mail Goggles, is explained in the Official Gmail Blog.
In summary, the Mail Goggles feature allows you to make Gmail aware of certain hours during the week when you should not be sending e-mail (due to exhaustion, inebriation, or the side effects of whatever other illicit things you do in your personal life). Once these hours are set, should you decide to send an e-mail during one of these highlighted times, you will first be prompted to answer a series of math questions, in an attempt to prove to Gmail that you have sufficient mental faculties to be sending e-mail.
A noble pursuit, to be sure. A trustworthy internet wingman may be just the thing for those among us who may enjoy their night life a bit too much, only to make decisions they regret in the morning. And while a wingman tied to your e-mail can’t help talk you down from every form of debauchery, there are certainly situations in which such a feature could be useful.
Unfortunately, Mail Goggles is kind of a fair weather wingman. He’ll check in on you every once in a while, but if you tell him you’re fine, he’ll leave you alone. He might tiptoe around the issue of whether or not you’ve had enough to drink, but talk forcefully enough and he’ll back down. That may sound ok, but sometimes you need a wingman who has the resolve to set boundaries for you when you’re not in a condition to set them yourself. Sadly, Mail Goggles is a bit too much of a pushover.
Maybe Mail Goggles could learn the laws of Wingmandom if it watched more VH1. Or if it bought a big poofy hat.
I have reached this conclusion after experiencing firsthand what the Mail Goggles system has to offer. Today I went in and warned my e-mail that between the hours of 3 and 4 pm on Wednesdays, I was not to be trusted with the “Send” button. I then attempted to send an e-mail to myself, and sure enough was prompted with a list of math questions.
Now, you can tell Mail Goggles how difficult to make the questions, by setting the difficulty to be a number between 1 and 5, 1 being the easiest, and 5 being, well, the less easiest. Not wanting to peak too soon, I asked for the easiest questions, and so was not surprised when the following problems greeted me:
10 x 6 = 32 + 18 = 85 – 10 = 10 + 10 = 95 – 85 =
I was slightly more surprised by the amount of time I had to solve these questions: 60 seconds. With a whopping 12 seconds allotted per question, even someone with minimal computational ability could easily plug and chug these answers through a calculator within the allotted time – and when you’re already at the computer, the thought certainly must be tempting.
And let’s be honest – you’d have to be pretty far gone not to know what 10 + 10 is.
Itching for more of a challenge, I went back into the settings and ratcheted up the difficult to level 5. Ready to get those synapses firing, I tried another test e-mail, and was given the following questions:
477 – 138 = 72 / 9 = 8 x 8 = 242 – 98 = 30 / 10 =
Again, Mail Goggles saw fit to give me 60 seconds to answer these questions.
Seriously? This is the difference between level 1 and level 5? You give me a couple of three digit numbers, and introduce the concept of division? Is this really the best we can do? Not to mention the fact that the calculator would still function as a perfectly good cheat sheet.
With a desire to test my limits, I answered one of the above questions in error, and stuck my hand out, waiting for retribution. But did any come? Sadly, no. Instead I was just given another 5 problems, and a full 60 seconds on the clock! Come on, Mail Goggles, where’s the accountability? If you can’t divide 30 by 10, maybe you should sleep on that email to your boss telling him how attractive you find his wife. But instead, Mail Goggles says to you, “Hey buddy, it’s ok. Just try again! You’ll get to that e-mail eventually, I know it!”
There are other issues I have with this innovation from Google Labs as well, but I don’t want this to turn into a negative tirade. The idea is quite inspired, but it leaves much to be desired, especially if you really want some checks in place before you do something you may not really want to do.
In the spirit of keeping things positive, to the designer of this feature, Jon Perlow, I humbly submit some suggestions for future improvements to Mail Goggles:
1. Make questions that aren’t so easy to answer with a calculator. How about more critical thinking questions? You can use calculators on the SAT, and because of that the questions are specifically designed so that the calculator may or may not be an asset.
2. How about some significant gradation between difficulties? If you’re going to differentiate between levels of mathematics, you might as well make the problems worthwhile to people with all kinds of backgrounds. I don’t think some calculus would be too much to ask in the higher levels, even dare I say it some linear algebra. At the very least, can we get a smidgen of long division?
3. How about instead of 5 really easy problems, you just give one or two problems that require more critical thinking? This will better test mental faculties – you can test me once with a hard problem, rather than testing the same thing 5 times with simple questions.
I know you mean well, Mail Goggles, but you’re really not looking out for people when they need you. For now, ladies and gentlemen, I suggest that you stay with your flesh and blood wingmen. They will make sure to keep you away from the computer as long as you are in their sight. They will protect you, watch over you, and make sure you do nothing unsightly.
Unless they secretly hate you, in which case they will probably take incriminating photos of you and post them on the internet. In this respect, Mail Goggles offers much more protection. For now.
Math made headlines again last week, with the announcement that the Program in Computing, a subset of UCLA’s Math department, had discovered a prime number with approximately 13 million digits. Among other places, this announcement could be seen on the front page of Yahoo! News – if you missed it, here’s the link. This discovery gives rise to some natural questions, which I will try to address here:
1) How big is a 13 million digit number?
2) Who cares about big primes?
3) Is this what mathematicians do all day?
The first question is the easiest to answer. In short, a 13 million digit number is very, very large. This link displays the beginning and ending string of digits that compose the newly discovered prime, and for those of you with some time on your hands it also has a link to the full text of the prime midway through the page (the page with the full printout of the prime takes up 16.73 megabytes).
To give another sense of perspective, suppose you read the digits in this prime out loud. Assuming you read aloud at a fairly steady pace (say, 2 digits per second) it would take you about 2 1/2 months to read through the entire prime (this is assuming, of course, that you do it all in one go, with no stopping for sleep, food, or other frivolities).
A pretty big prime. Just not as big as the one discovered at UCLA.
As for the second question, why should anyone care that a group of people found a really big prime number? Of course, it is well known that there are infinitely many prime numbers, so it’s not unexpected that there exist really big ones. Given that there are infinitely many, why should we care about finding specific ones? Isn’t it a little bit like trying to see how far you can send an astronaut into outer space? What’s the application?
Well, there isn’t much. The most significant application of prime numbers is to cryptography, more specifically, public key cryptography (those interested can begin to further investigate this connection here). Other than that, though, there really is little in the way of applications for the discovery of large primes. Enthusiasts will say that the search for large primes is akin to the quest to climb Mount Everest, or the desire put people on Mars. Such people, when asked why we should look for large numbers, may appropriate the words of George Mallory and reply, “Because they are there.”
However, while the search itself may not have applications, the techniques used to aid in the search often do. It has certainly been the case many times in mathematics that mathematicians will develop a theory with no concern for any practical application, and it may not be until several years or decades later that an application emerges for the theory which has been developed. Perhaps those of you who are not enthusiasts can take solace in this fact.
A summary of other common reasons why people may search for large primes can be found here.
As for the last question, the answer is a resounding “no!” – this is not what mathematicians do all day. There is a FAQ on the discovery of this latest prime, written by Edson Smith, who headed up UCLA’s quest. At the beginning of this FAQ, he makes sure to point out that he is a systems administrator and not a mathematician – and indeed, this result may be of more interest to computer scientists than mathematicians, because of the computational methods used to discover this prime. Or maybe not – not being a computer scientist, I can’t say.
Whatever the case, mathematicians do not spend their days looking for prime numbers. The field has infinite breadth and infinite depth, with prime numbers taking up a small section, and the search for large prime numbers taking up an even smaller subsection. This is not to downplay the achievement made, but it is important to note that much of what is reported about mathematicians in the news bears little resemblance to what mathematicians actually do.
Prime numbers are something everyone learns about in grade school, and for that reason a result such as this generates excitement because it involves current work, but it can be explained to a non-mathematician very easily. For this reason, it may seem like the general population is getting an insight into the secret work that we do. However, this work bears very little resemblance to the everyday research of most mathematicians.
As an achievement in computing, it is certainly worthy of applause. But for those of you who think you have learned just what it is we mathematicians are doing, I’m afraid this result gives no indication of how far the rabbit hole goes. To get a sense for this, I would encourage you all to take more math classes. It is never too late to learn more math!
As you may recall, I have already discussed certain perils associated with different voting systems. However, given all the commotion this election is causing, I thought it may be worthwhile to discuss voting in a bit more detail.
There is plenty of information online regarding the relationship between math and voting, for those with interest enough to seek it out. But perhaps the best centralized internet location on this topic comes from this year’s Mathematics Awareness Month website.
In April of every year, mathaware.org hosts a Mathematics Awareness month, complete with articles and contests related to the year’s theme of forging a bridge between mathematics and what is often times a seemingly disparate discipline. It was no doubt with tremendous foresight that they selected “Mathematics and Voting” for this year’s theme.
A good way to kill a few minutes is with their voting methods simulation. On this page, you can vote for potential presidential candidates using three voting rules. The first voting rule is the standard: vote for a candidate. The second rule allows you to cast a vote for as many candidates as you wish. In the third rule, you are asked to rank your candidates from first to last, and points are then awarded to each candidate according to the Borda Count – essentially, if you are choosing between n candidates, the Borda count awards your first choice n – 1 points, your second choice n – 2 points, and so on, so that your last choice receives 0 points.
I highly recommend you vote for yourself. Once you do so, you can view the results, which are quite interesting in and of themselves.
Beyond that, there are several interesting articles you can peruse at your leisure by following this link. I don’t have the time to discuss every article, but I will highlight a couple here.
The first, in very literally keeping with the theme, is called “Mathematics and Voting,” by Professor Donald G. Saari. This article talks in detail about the problems with voting systems, which include the three given in the simulation discussed above. All the voting systems under consideration fall prey to Arrow’s theorem (discussed in the last post), but even so, it is natural to ask: are some voting systems more broken than others? The short answer is yes.
Professor Saari not only discusses how bad things can become (for instance: given three candidates to choose between there is a roughly 69% chance that the choice of voting rule can alter the outcome of a close election), he discusses how and why some voting rules are “better” than others (by “better” I mean that some voting rules do not allow as many undesirable election outcomes as others). In fact, the aforementioned Borda count allows for far fewer potential election outcomes than most other ranking systems.
Does this suggest that in some sense, the Borda count is better than other voting rules? It is even more interesting to ask this question in light of the simulation election results one can view after voting.
Another paper (which I highlight because it is such a short read) is titled “Yes, It Is Rational to Vote” by Professor Andrew Gelman. He offers an interesting rebuttal to the commonly held view that it is irrational to vote, because the chances of your vote actually determining the outcome of an election are so minimal. His argument is that even if you believe your preferred candidate will bring a marginal increase in the quality of life to the citizenry (equivalent to some small monetary sum, say $50), when we multiply this montary amount by the population of America, the expected value of your vote becomes non negligible. Although the odds of your vote being decisive are extremely small (especially in red state or blue state strongholds), a small benefit compounded by 300 million people then translates into a significant return.
There are plenty of other articles as well, and while some are more technical than others, certainly all of them merit a look.
I would like to close by pointing out that all of the voting systems so far discussed have been based on a ranking system. Given candidates A, B, and C, you are asked to order your preferences, and every voting rule is an attempt to quantitatively translate your preference.
However, just giving a preference does not give the whole story. For example, you and I may both prefer A over B and B over C, but you may love A and hate C, while I feel relatively indifferent among the three candidates. In this case, although we would both say that our rankings are A > B > C, this simple ranking ignores crucial information about the degree of our preferences.
Next time, I will look at the concept of Range voting, which attempts to address this issue. One interesting feature of range voting is that it satisfies all the conditions of a voting system one would like to have, in what may seem like a contradiction to Arrow’s Theorem. It is, however, because of the added information put in to the range voting model that Arrow’s Theorem is not violated. We will discuss this in more detail, and look at perhaps the most common example of range voting, next time.
