There is a joke in mathematics circles that has become well-known enough to merit its own entry on Wikipedia. This joke is referred to as the Teakettle Principle. Here’s how it goes:
A mathematician and an engineer go into the kitchen one day to make a pot of tea. Finding an empty kettle on the stove, they fill it with water, then turn on the stove and let the water boil, following the usual protocol when making tea.
The next day, the two again decide to make a pot of tea. However, upon entering the kitchen, they find that the kettle on the stove has already been filled with water! Now faced with a new problem, the engineer suggests that they simply heat the water that’s already in the kettle.
“Nonsense!” the mathematician replies. “It would be far simpler to pour the water out and replace the empty kettle on the stove top. Then we will have reduced this problem to one we have already solved!”
Ok, so the joke isn’t that funny. Its charm comes from its needling of the mathematical proclivity to reduce new problems to ones that have already been solved, sometimes even when a solution to the new problem may be readily available.
It is with this mentality that I wish to discuss Superman II with you. For you see, the problem of discussing this movie as it pertains to mathematics has already been solved in an excellent post at Overthinkingit. The name of the post is called The Math of Steel, and it thoroughly analyzes, from a mathematical perspective, the plausibility of a scene involving Superman, Niagara Falls, and a falling child. It’s really a very good example of how even superheros can benefit from a knowledge of mathematics.
Therefore, since this problem has already been solved, the problem of me discussing this film from a mathematical perspective can now also be solved, just by using the link above. You can thank the teakettle principle for that.
Ah, 1993. Andrew Wiles was on the verge of proving Fermat’s Last Theorem. Late night talk show hosts poked fun at our President’s love of McDonald’s. And on June 11th, a little film known as Jurassic Park released to audiences throughout the country.
As it held the top spot for most successful movie of all time for four years (thank you, Titanic), there is no doubt this movie has secured a place in our pop culture heritage. And while it has aged in some respects – science has advanced to the point where it can genetically engineer species that went extinct millions of years ago, but a little girl is still most impressed by the fact that cars on the island come equipped with “interactive CD-ROMs,” for instance – the film still serves up a quintessential example of the 90s summer blockbuster.
If the film is not fresh in your minds, you may be asking yourself what a movie about dinosaurs wreaking modern day havoc has to do with mathematics. In response to this, I turn your attention to the character of Dr. Ian Malcolm, played by Jeff Goldblum. Dr. Ian Malcolm is a mathematician, although he self-importantly refers to himself as a “Chaotician,” i.e. his research is in Chaos Theory.
Putting that aside for a moment, let us take a look at this portrayal of a mathematician. Does the film do a disservice to those of us in the profession, or does it raise us up, so that we can walk on mountains?
If you have lived under a rock for the past 15 years, here is a (poor quality) trailer for the film.
Let’s take a look at some of these stereotypes.
- People who are good at math are socially awkward. While Dr. Malcolm certainly has his fair share of idiosyncrasies, I believe a rational person would find it difficult to label him socially awkward. During the course of the film, he proves himself to be quite a conversationalist, and is not shy about voicing his opinions, or interacting with people he has just met.
Ian Malcolm with his stunner shades on.
Not only is Dr. Malcolm able to hold his own in a conversation, but he is also way more stylish than most portrayals of mathematicians in pop culture. Sporting both a pair of sunglasses and a leather jacket, there is no doubt that Dr. Malcolm has a keen eye for fashion, and the means to support his tastes.
Perhaps most importantly, at one point the owner of the park, John Hammond, refers to Dr. Malcolm in the following way when discussing personnel that have been brought to the island: “I bring it scientists; you bring a rock star.” While I believe this is meant to be an insult, it is more constructive to interpret this as an affirmation of the inevitability of a future utopian society in which mathematicians are given the adoration reserved today only for rock stars. The film captures the essence of this utopia quite nicely, except for the bit about the dinosaurs running around and eating people; this is not (thus far) a part of the vision.
In short, Dr. Malcolm is quite far from being socially awkward. +1.
- Male mathematicians have a crippling fear of talking to women. A strong argument could be made that this stereotype is really more of a subset of the stereotype already discussed, but because of the emphasis this film places on Dr. Malcolm’s gift of gab with women, I feel it is worth mentioning here.
In several scenes we witness how Dr. Malcolm has no reservations about spitting some serious game to females, even in front of other people. Rather than letting mathematics restrict his ability to talk to the opposite sex, Dr. Malcolm uses mathematics as an opening to get women to talk with him. In doing so, he illustrates one of the greatest unsung properties of mathematics: when used with the proper care, it is a powerful aphrodisiac. Kudos to you, Jurassic Park, for daring to shed light on this important facet of mathematics. +1.
Dr. Malcolm is looking for love.
Now, what about the bad? Unfortunately, there is plenty of bad, most of it coming from taking the good stuff too far. Let’s look back at the stereotypes already mentioned.
- People who are good at math are socially awkward. Sure, Dr. Malcolm may be socially adept, but he’s also kind of a jerk. I say this not because of his criticisms of the park, many of which seem quite valid. Instead, I base this conclusion on the fact that he’s a little full of himself. From the fact that he wears sunglasses at night (an action that would be excusable given an appropriate medical condition, but since he later loses his sunglasses, we can safely assume this is not the case), to the way he refers himself as a “Chaotician” (come on, seriously?), it is clear that Dr. Malcolm is too pompous to serve as a proper mathematics ambassador to the rest of the world.
So abrasive is this mathematician that at one point, one character refers to him by saying, “I really hate that man.” Is this what we want people to say about mathematicians? Perhaps, but only because they envy us. Certainly this does not seem to be the case here. -1.
- Male mathematicians have a crippling fear of talking to women. Although Dr. Malcolm evidences an ability to use mathematics in his courtship rituals, this commendable feat is overshadowed by the fact that he uses his forces for evil, and not for good. The only time we see him channeling the mack within is when he is putting the moves on somebody else’s girlfriend. This is, of course, a universal party foul, and one that does not reflect well upon mathematicians. Especially when the other man is a more likable character. -1.
In summary, this film really doesn’t do much for the perception of mathematicians one way or the other. While Dr. Malcolm is more suave and sophisticated than most people come to expect from their mathematicians, he is also in love with the sound of his own voice, and potentially a home wrecker. In the end, some may be able to look past the character’s shortcomings, and some may not. However, I suppose that any man who can serve as the inspiration for the following video can’t be all bad.
Does the situation improve in the 1997 follow-up, The Lost World? Perhaps, although I don’t really want to sit through the film to find out.
Recently, I found myself thinking of mathematics in an unlikely set of circumstances: while watching VH1’s latest “Celebreality” show, Brooke Knows Best. I realize that an admission like this may be embarrassing, and so it is for the sake of your edification, dear reader, that I am willing to go on the record with this deliciously shameful information.
For those of you who may not know, the titular character is the daughter of Santa with Muscles star and All-American hero, Hulk Hogan. In the show, Brooke lives in an expensive looking condo in Miami, goes to the beach, and sings her own theme song. This is about as much as I know. I swear. For those of you who are curious, the following video gives a good sense of what this show is all about.
The particular episode to which I would like to draw your attention (Episode 4) involved Brooke traveling to Panama City, Florida, to host some Spring Break parties. And I bet you thought math and spring break were incompatible.
Because of his overprotective nature (or because the producers thought it would make better television), Hulk Hogan decides that it’s a good idea to tag along on these spring break adventures. He even brings his chubby, mullet-sporting friend in tow, and the two of them cause all kinds of PG-13 hilarity. It is as if they are the love children of the love children of Abbott & Costello and Schwarzenegger & Stallone.
Hulk Hogan sees dead people.
One day, Hulk suggests that the group go visit The World’s Largest Human Maze (or, as he calls it, “The World’s Largest Human Maze in the world”). The maze in question is the Gran Maze of Panama City. Calling it the World’s Largest Human Maze is a bit deceptive, as there are hedge mazes which are larger – however, to the maze’s credit, it does not seem to refer to itself as the World’s Largest Maze in any of its information.
In any event, they go to this maze: Hulk and his buddy as one team, along with Brooke and her two roommates as another. At some point during their travels through the maze, they decide to split up and make a competition out of it: whichever team can make it through the maze first will get to plan the agenda for the rest of the day. Hulk and his friend make it through first, but only by squirming underneath the wall panels. The other team, once they learn of this deception, declares Hulk’s victory null and void.
Hulk Hogan: setting a bad example for maze solvers worldwide.
There is certainly a moral here: don’t cheat. But there is another important moral here, one I think may be lost on Hulk Hogan, even to this day. That moral is this: Hulk, you should have brushed up on your maze-solving algorithms!
Indeed, there are a number of procedures one can follow in order to try and solve a maze. Perhaps the most well known is the Wall Follower algorithm. In this procedure, you walk through the maze keeping either your left or right hand in contact with the wall of the maze at all times. If you do this, without removing your hand from the wall, you will eventually find the exit. Here is an example:
Solving a Maze: The Wall Follower algorithm. This maze was constructed using the Maze Maker.
Now, the astute reader may realize that unfortunately, this method will not always lead you to the end of the maze. Even in a simple example, such as the one below, it is easy to see that regardless of whether you follow your left hand or your right hand, you will merely circle the entrance and never reach the end.
Failure of the Wall Follower algorithm
So, how do we know if the Wall Follower algorithm will work? This algorithm will always lead us to the exit (provided there is one) as long as the maze itself is simply connected. Less rigorously, when the maze cannot be solved using this algorithm, it is because the maze is in separate pieces. In the example above, we see that the piece of the maze that surrounds the start is disjoint from the rest of the maze.
Why, then, does this algorithm work if the maze is simply connected? Because every simply connected maze can be continuously deformed to a circle (i.e. such mazes are homeomorphic to the circle). Once we deform the maze to a circle, it is obvious how the wall follower algorithm works: it is equivalent to simply tracing your way along a circle between two points. This video does a better job explaining what goes on:
So, had Hulk confirmed that this human maze was, in fact, simply connected, he could have used this algorithm to try and beat his daughter fair and square. However, what if the maze is not simply connected? Indeed, a zoomed in view of the maze, courtesy of Google Maps, is somewhat inconclusive. How then, could Hulk guarantee that he would eventually make his way out?
For more general mazes, we can turn to another algorithm: Tremaux’s algorithm. This algorithm will lead you to the exit even in mazes that are not simply connected. However, Tremaux’s algorithm also requires something that the Wall Follower algorithm does not: namely, you must have some way of marking your path.
The algorithm can be described according to the following rules:
Walk down the maze, drawing a line behind you.
When you come to the first intersection, choose a path at random and follow it.
When you come to a dead end, turn around and return to the last intersection.
If you are walking down a corridor that you have not been down before, and you come to an intersection you have already visited, treat the intersection as a dead end and turn around.
If you are walking down a corridor that you have been down before, and you come to an intersection (necessarily one you have already visited), then go down a path which you have not visited yet, if possible. If this is not possible, go down a path you have only been down once.
Rule 5 may look restrictive, but in fact, with this algorithm you will never need to walk down the same corridor more than twice. If the maze has a solution, this algorithm will find it. Moreover, once you have used this algorithm to find a solution, the corridors marked only once will trace out a direct path to the finish.
Solving a Maze: Tremaux’s algorithm. Notice that no corridor is traveled more than twice, and the ones traveled exactly once trace a direct path to the finish.
So there you go, Hulk. Two methods you could’ve used to try and defeat your daughter fair and square. Granted, there is no guarantee that following these procedures would have gotten you through any faster than your daughter, but they certainly wouldn’t have disqualified you from victory. Instead of being able to savor your win, you were called a cheater, and your prize was revoked.
Let this episode serve as a lesson to aspiring professional wrestlers everywhere: math can be of service, even to you.
For more on maze algorithms (both algorithms for creating mazes, or for solving them), this website offers a treasure trove of useful information. Hopefully Hulk Hogan will do his research the next time he makes such a mathematical challenge.
Hulk Hogan is ready to listen to what mathematics has to say. Are you?
As you may recall, my first post briefly discussed the California Board of Education’s mandate that every 8th grader in the state must take Algebra. My purpose here is not to discuss the ruling further, but rather to point out the response article published last month in the San Francisco Chronicle.
The article is well-researched and thoroughly written. Not only does it feature discussion of the pros and cons of such a mandate from a wide range of interviewees, but it also tries to address the question of why Algebra, and mathematics in general, is perceived so terribly by American kids and adults alike. It also attempts to paint a picture of what Algebra actually is, for those of us who fell by the wayside of mathematics long ago.
The current state of mathematics education is given quite a scathing review by the people mentioned in the article who actually know their mathematics. The harshest critic is former UC Santa Cruz mathematician Paul Lockhart, who wrote the following in a 2002 essay:
If I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done … I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.
What is it about contemporary mathematics education that is so broken? There are three factors discussed in the article, and all of them are spot on: the failure of our curriculum to make math seem relevant, the educational system’s focus on testing, and the lack of qualified teachers.
Even though it is 2008, and technology surrounds us, many students have trouble seeing how math will help them later in life. Given the exponential growth in technological industries, not to mention reliance on data and statistics that is prevalent throughout the social sciences and other careers such as medicine, it is unlikely that this lack of foresight is due to the dwindling relevance of mathematics. If kids don’t see why math is useful, it’s because we’re not doing a good job showing them.
The article gives some examples of how math is used in unlikely places – from parking cops to delivery trucks to iPods. The main argument behind these examples is “Hey! Algebra is relevant – look at all this cool stuff that uses it!” On one hand, this may seem a bit deceptive. After all, saying that all you need is some Algebra to understand the machinery behind an iPod is a bit like saying that all you need to become a French pastry chef is knowledge of how an oven works – both make oversimplifications of the knowledge required. But on the other hand, algebra is a vital piece of foundation you will need in order to understand many modern technologies, even if a complete understanding requires a much deeper understanding of mathematics. It may be difficult to explain all the intricacies of how a circuit board works to a fourth grader, but saying something is better than nothing – especially when that fourth grader believes that studying math is essentially pointless.
“Math? My iPod taught me everything I need to know!”
The aforementioned Professor Lockhart advocates a less rigid approach to mathematics education, one in which the students can more freely explore mathematical ideas. Perhaps the reasoning here is to help guide the student down the same path of discovery that first led to the concept being taught. This is indeed a good way to teach mathematics, because it intertwines the concept with our own intuition, so that rather than seeming abstract and separate from reality, mathematics is seen as a way of interpreting the world in a rigorous, but natural way.
Unfortunately, such a pedagogical approach does not blend well with that pillar of American education: standardized testing. When teachers are forced to teach to a test, the motivation for studying mathematics is no longer to achieve a deeper and richer understanding of the world, it’s to fill in bubbles with a No. 2 pencil quickly and with minimal error. This brings up a slew of other issues: for example, how can we be sure that the tests are actually testing the mathematical knowledge we want the students to acquire? The article gives some examples of test questions at the end, but all of them can be solved by simply checking the given answers. While not an efficient test taking strategy for every question, it can certainly be used often enough to give the impression of mathematical competency.
Finally, the article pointed to a somewhat startling result: about a third of middle school Algebra I teachers do not have a math credential, and given the algebra mandate, that number is only expected to go up. Of course, there are probably good algebra teachers around who may not have a math credential, but at the same time, there are many math teachers in this country who are underqualified. Sadly, Jaime Escalante is but one man, and can only reach so many kids.
Given all this, the state of math education in this country may seem dismal. Perhaps it is. Will anything be done about it? I sure don’t know. But I am curious to see what effect, if any, this mandate will have. If nothing else, it should make for some interesting discussion.
For every guy who has dreamed of looking like a Hoobastank concert attendee, or for every girl who has dreamed of looking like a Bratz doll, Yahoo! Answers provides you with a forum to not only construct the avatar of your dreams, but also to ask questions on a variety of topics, and get real answers from real people.
She only loved him for his soul patch.
Unfortunately, as a math educator, I feel compelled to offer criticism regarding the Mathematics section of your site. The existence of this section is not what bothers me – it is the user behavior, both of those asking questions and those answering them.
Let’s start with the askers. I’ll be blunt: please stop using the internet as your interactive cheat sheet. There are a number of users of Yahoo!’s service who have no qualms asking for answers on their homework – many do it blatantly, and with no regard to the importance of these formative years in their mathematics education.
No attempt is made to give any sort of context: what part of these problems is difficult for this individual? Moreover, since these problems are all solved in exactly the same way, it is clear this person is only looking for quick answers to their homework, and not an understanding of how to solve these types of problems.
Of course, these are some of the most obvious offenders, but there are numerous other examples of students (mostly younger ones) abusing the privileges given to them by the folks at Yahoo!.
Needless to say, this is something that should be discouraged. Parents taking a more active role in monitoring their child’s internet habits will help, of course. But it is just as important to change the behavior of those answering the questions, because no matter how obvious it is that you are trying to cheat, somebody will answer you, with no questions asked.
In the case of Exhibit A, as of the time of this writing there are 16 answers to the “question.” Of these, 8 give answers to all nine parts of the question, and of those 8, 3 give answers without showing any work whatsoever (note that giving an answer is not the same as giving a correct answer). My favorite response is provided by a gentleman who chastises the questioner by saying, “It’s not fair to use Yahoo! Answers to solve your maths homeworks,” and then proceeds to write down the answer to every question.
There is only one answer to exhibit B, but sure enough, it is a link to a website that will do all the work typically asked of a middle school student in math. Not a word of reprimand is offered.
We have all heard the old adage praising the benefits of teaching someone to fish, rather than simply giving someone a fish. The people who answer these types of questions, however, don’t just give you the fish: they clean it, cook it, serve it, cut it up into tiny pieces, then chew it for you and spit it into your mouth. All that’s left for you to do is receive the regurgitation, like baby birds. The difference being that baby birds don’t need to know how to add fractions.
The thought of his daughter never learning how to fish is enough to make this man cringe.
What can be done? While mathematical abuses at all levels of education seem to be occurring on Yahoo! Answers, it is more prevalent with the most basic material, suggesting the most common culprits are the very students who most need to build a mathematical foundation. For more advanced levels or mathematics, I shift the blame more towards the questioner, but for younger students, who may not see what long term harm a lapse in their mathematical understanding can have, the onus moves to those answering the questions, who should know better than to sell out their knowledge for some measly Yahoo! Answers “points” (the purpose of which is still not entirely clear to me).
It is with this in mind that I implore you, Yahoo! Answers users, to stop spoon feeding middle school homework answers to students. You do them a disservice, and by failing them, you are failing our collective future.
I should mention that not all of the mathematics category is corrupted. There are plenty of examples of people asking intelligent questions (questions aimed at understanding material and not just getting the right answer, for instance), and there are also many examples of people answering the questions only partially, but giving full details for the parts they do answer so as to help guide the student’s understanding. This is all well and good. But as long as someone can get answers to their homework without having to think, our work is not complete.
In conclusion, letting the internet do your homework for you is most likely a bad idea. Using the internet to help you understand your homework is, in general, a good idea. So parents, make sure you know whether your child is using the internet for good or for evil. The distinction could cost them an understanding of mathematics – or even their lives.
Well, maybe not their lives. But certainly this behavior should be discouraged.
He’s coming for you, disbelievers
