Math in the News: Counting without Language

Last week, some of you may have seen this article about a study on Australian aboriginies. The study suggests that, even without having the language to describe numbers, the human mind has an innate ability to count and differentiate between numbers.

Australian Aboriginies: Math All Stars?
The study focused on two Aborigine tribes in Australia, and found that even though both tribes lack words for individual numbers (the languages only have words to describe ‘one,’ ‘two,’ ‘few,’ and ‘many’), members of the tribe nevertheless seem to have a sense for different numbers and counting. This conclusion was reached, for example, by banging two sticks together n times, and asking children to represent those n times with concrete objects.

I am no linguist, so I cannot speak to the linguistic ramifications of this study. From a mathematical viewpoint, however, it is certainly a good thing to hear, because it suggests that the ability to count is innate within all of us, independent (in a sense) of processes so natural that we take them for granted (e.g. language).

It is tempting to extrapolate broad claims from this, e.g. mathematics is so fundamental, we all have an ingrained understanding of it on a basic level. This may sound a bit insulting, however, to those who have struggled in their math career, and is also a bit unfair since we can’t really extrapolate much about mathematics in general on the basis of this one study. However, the suggestion that such a fundamental mathematics principal is somehow innately wired into our brain hopefully will help persuade skeptics that, at least on some level, mathematics is not a black box – in fact, the theory is often developed from natural and intuitive principles, and is motivated by the desire to solve real-world problems.
At the same time, one hopes this news does not spell the end for Count von Count. Even though we apparently can count without the language to describe what we are doing, having the language certainly can’t be a bad thing, can it? For his sake, let’s hope not.

Count von Count may need to diversify.

August 26, 2008 | 4 Comments »

Math in the Movies: Good Will Hunting

Winning them the Oscar for Best Original Screenplay in 1998, Good Will Hunting propelled Matt Damon and Ben Affleck to the Hollywood A-list (no doubt Phantoms would have done this for Ben Affleck, had it not been for the success of Good Will Hunting only months earlier). I will not summarize the plot, except to say that in this film, Matt Damon is a math superstar. For those wanting more in the way of plot summary, this trailer may help to refresh your memory:


There are a number of films that center around math geniuses, and for the most part they have met with some degree of critical and commercial success. Our purpose here is not to critique these films, but to answer a simple question: In what ways do these films perpetuate stereotypes about mathematics and mathematicians, and in what ways do these films rise above those same stereotypes? Here we use a rather crude scoring system, whereby each perpetuated stereotype corresponds to a loss of one point, while each stereotype that is overturned corresponds to a gain of one point.

Let’s start with the good.

- People who are good at math are socially awkward.
Will Hunting certainly runs counter to this statement. He has several close friends, enjoys going out for drinks, and lands a smoking British girlfriend. Will also really likes punching people (see below), a pastime I think it’s safe to say is not shared by most people who study math.


Watch out! This clip, and the one below, may contain objectionable language.

While Will Hunting is not socially awkward, not everyone in the film can escape the same fate. Fields medal winning, ascot adoring math professor Gerald Lambeau seems relatively well adjusted, but his teaching assistant Tom is painfully awkward – almost awkward enough to overshadow the non-awkwardness of the rest of the mathematically inclined cast. Perhaps he simply got lost in Matt Damon’s eyes.

Overall, the film does a good job of showing that not everyone who is good at math is a social outcast. +1.

- Mathematicians are single-minded of purpose, and have no interest in anything besides mathematics.
Will Hunting provides a strong counter-example for this assertion as well. The movie shows us that Will is not just a math prodigy, but is basically an all-around smart dude. Very little time is spent showing the audience how Will does math – understandably so, since I can imagine few things less cinematic than watching someone think about math problems. Because of this, we see the other aspects of his character, and learn that his realm of interests extends far beyond what can be found in a collection of math books. +1.

Among other activities, Will enjoys schooling Harvard punks.
He also enjoys eating caramels.

- To be good at math, you must be insane.
While he certainly has some issues, I don’t think anyone would assert that Will has any significant mental disorders. This film shows that you don’t have to be crazy to enjoy math – although it couldn’t hurt, I suppose. +1.

What about the not so good?

- It is important to do math on windows, mirrors, or other unconventional writing surfaces.
I get it. Watching people sit at a desk and write things down is boring. Nobody wants to watch that. But if you go to a mathematician’s office, I doubt you will find writing on the window, or on the mirror of their bathroom at home. That’s what blackboards are for. -1.

Dude must’ve run out of chalk.

Other than that, the film does a fairly good job. It is understandable vague when it comes to the supposedly insanely difficult problems Prof. Lambeau gives his students throughout the film, although the brevity of their solutions seems to suggest that really they weren’t so bad in the first place. The film does earn points, however, for featuring Parseval’s identity in the first classroom scene.

In conclusion, the film earns two points here, for its portrayal of math students as real people with varied interests, even if they do enjoy writing equations on non standard surfaces. One wonders if the sequel will be able to compete.

August 18, 2008 | 2 Comments »

Math Gets Around: Dating

Do you wonder whether you will ever find true love? Are you tired of looking for Mr. or Ms. Right? (I mean this in a metaphorical sense – if you are actually looking for an individual by the name of Right, this article will probably be of no use to you.) Have you grown weary of idle party chit-chat, and awkward mornings after nights spent in venues with deceptive lighting? Well, my friend, whether you are willing to accept it or not, mathematics can help you find the one to share your life with.

Unfortunately, the primary disadvantage to the method described below is that if you don’t know about it before you jump into the dating scene, it may be too late for you to utilize. But with an open mind, and a willingness to let mathematics do its work, you can maximize the likelihood that you will find that special someone.

How is it that math nerds get all the girls? Read on to find out!
(AP Photo/HO/20th Century Fox)

How can mathematics help you find a mate? Let’s formalize the search for love as follows: you want to know when you should commit. Usually the way it works is that you date a few people until you find one with whom you are compatible enough, and one who does not drive you too crazy, and hopefully one for whom these feelings are mutual. So, you date until you find someone with whom you are reasonably confident you could settle down.
But how do you know that someone better won’t come along? Indeed, this is a question that may plague those afraid of commitment. Or, for those of you who have been in the game for some time, you may be asking: when should I throw in the towel and settle down with Mr. or Ms. “I guess they’ll fit the bill”?
If you listen to the advice of mathematics, you will do the following:
Date roughly 37% of the total number of people available to you in your dating pool, but do not stay with any of them. After dating this first 37%, stick with the first person that comes along who is better than any of the previous candidates.1
If you follow this process, you will pick your ideal mate from out of your dating pool 37% of the time – this is the highest probability you can get, given these circumstances. This is also assuming that there is no going back in your relationships: once things are done, they are done for good.

Had they known that backpedaling would’ve hurt their chances of finding
true love, perhaps Ennis and Jack could have saved themselves some heartache.
Even cowboys can benefit from a knowledge of mathematics.

Let’s take an example. Say that you plan on dating 100 people during your life. In order to find the best match for you, you should date the first 37 people, but not settle down with any of them. Then, the first person who comes along after those initial 37 who you like better than anyone you have dated before is the one you should stick with.

Of course, there are some natural questions that this process raises. Most importantly, what exactly qualifies as “dating”? The nice thing is that however you choose to define dating, this process still applies. So in fact, the choice of definition is not so important. All that matters is that you follow this process according to your definition.

Other common questions fall along the lines of: What if I pass up my true love in that initial 37%? Or what if my true love is at the very end, and so I pick someone else before meeting him or her? Indeed, these are valid concerns. However, as with many things in life, the quest for love is not easy, and there is no guaranteed way to find the one you are looking for. As stated above, this method will provide you with the highest probability of success, but even so, roughly 2/3 of the time you will wind up with a sub-optimal match. Isn’t it romantic?

There are some valid criticisms to this method. One is that few people set out on their dating path with a set number of people they would like to date in their lifetime. Without this number firmly in your mind, how will you know when you have passed the 37% threshold? Because of this, for many people it may be too late to maximize their chances of finding true love.

Another criticism is that in these modern times, the dating game is less information-blind. The above description of dating assumes that you know nothing about your potential dating partners until you actually date them – in particular, you have no idea how compatible you may be with a member of your dating pool until you have begun dating. With the rise of internet dating, however, this is no longer the case: for example, match.com uses sophisticated algorithms to process user data in an attempt to predict compatibility between two people. Therefore, when you go to such a site to try and find a date, you have a sense beforehand of how you will fit with your potential partners: in effect, this allows you to see the ranking beforehand, so that picking should become easier.

The truth, as we all know, is that no method is foolproof. Even if you decry the 37% rule for dating, however, you may find it come in handy in other aspects of your life. What about when you are looking for a new apartment? Or trying to find that new job? In both these cases, there is usually no going back once you have looked at a potential candidate, and so it is natural to ask how long you should look before committing. The rule still holds: you maximize the chance that you are picking the best outcome by going through the first 37% of all available options, and then picking the option that is better than anything you have previously seen.

Note that as a corollary to this rule, as long as you are planning to consider at least two options, you should never pick the first thing you see. This goes for jobs, apartments, pets, cars, and so on. Sorry, high school sweethearts.

Sadly, those kids from High School Musical are screwed.

Given that this method is the best way for you to find the best match, is it how people act in everyday life? Sadly, the answer is no. The Wikipedia entry on this problem discusses some experimental studies on how compatible this algorithm is with real world behavior. In general, research has concluded the following:

In large part, this work has shown that people tend to stop searching too soon. This may be explained, at least in part, by the cost of evaluating candidates. Extrapolating to real world settings, this might suggest that people do not search enough whenever they are faced with problems where the decision alternatives are encountered sequentially. For example, when trying to decide at which gas station to stop for gas, people might not search enough before stopping. If true, then they would tend to pay more for gas than they might had they searched longer. The same may be true when people search online for airline tickets, say.

The lesson is clear: patience is a virtue. Don’t sell yourself short after examining 10, 20, 0r 30 percent of your potential candidates. Hit that 37% with confidence! In the long run, it will save you money, frustration, and heartache.

With greater knowledge of mathematics, bands such as this would not
have to be so emo.

1. 37% may seem like an arbitrary number, but the percent is actually 1/e, which is approximately 0.367879… . Wondering how e appears in this problem? Try to derive the solution for yourself! Or, if you have little background in probability, you can find the solution online, with enough sleuthing.

August 13, 2008 | 3 Comments »

Math in the News: Monty Hall Strikes Again

I apologize in advance for the fact that this references an article that is four months old. However, given the connection between the Monty Hall problem and popular culture, it cannot rightly be overlooked here, and this article from the New York Times allows us to discuss this problem from a unique perspective.

The Monty Hall problem is so named because of its origins in the game show “Let’s Make a Deal.” The problem itself is famous for having a completely counterintuitive solution, and my goal after discussing the problem and its relationship to the New York Times article on cognitive dissonance will be to explain where this disconnect between the problem and our intuition arises.

Here is a rigorous and unambiguous statement of the problem:

Suppose you’re on a game show and you’re given the choice of three doors. Behind one door is a car; behind the others, goats. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one randomly. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you “Do you want to switch to Door Number 2?” Is it to your advantage to change your choice?1

Of course, those of you familiar with the problem already know its solution: you are twice as likely to win a car if you switch doors.

There are many arguments for why this is true, but for the sake of brevity let’s only look at one. Assume that you pick door number 1 (the odds will be the same if you pick door 2 or 3). If the car is behind door 1, then clearly you will lose by switching. However, if the car is behind door 2, Monty Hall is forced to show you what is behind door 3, and so by switching you will win. By an analogous argument you will win if the car is behind door 3. Thus, if you switch, you will win if the prize is behind door 2 or door 3, and lose if the prize is behind door 1. Since there is a 1/3 chance the prize is behind door 1, and a 2/3 chance the door is behind door 2 or 3, we see that it is always in your best interest to switch.

The New York Times article linked above shows this counterintuitive problem is important to understand not just if you are a fan of probability – the problem can arise in seemingly unrelated fields. In particular, it seems to be at the heart of a logical fallacy in a classic psychological experiment.

Here is a description of the experiment, involving M&Ms and monkeys, as taken from the article linked above:

For half a century, experimenters have been using what’s called the free-choice paradigm to test our tendency to rationalize decisions. This tendency has been reported hundreds of times and detected even in animals…

The Yale psychologists first measured monkeys’ preferences by observing how quickly each monkey sought out different colors of M&Ms. After identifying three colors preferred about equally by a monkey — say, red, blue and green — the researchers gave the monkey a choice between two of them.

If the monkey chose, say, red over blue, it was next given a choice between blue and green. Nearly two-thirds of the time it rejected blue in favor of green, which seemed to jibe with the theory of choice rationalization: Once we reject something, we tell ourselves we never liked it anyway (and thereby spare ourselves the painfully dissonant thought that we made the wrong choice).

But Dr. Chen says that the monkey’s distaste for blue can be completely explained with statistics alone.

How does this relate to the Monty Hall problem? The monkey’s stated preference of red over blue is analogous to Monty’s choice of which door to open. Once the monkey shows that he prefers red to blue, there is a 2/3 chance he prefers green to blue as well.

Monkeys love red candy.

To see why this is true, let’s list out all possible orderings of the monkey’s M&M preferences. We already know that red is preferred to blue, so this eliminates some potential orderings (it brings the total number down from 6 to 3). The remaining choices are:

Red > Blue > Green
Red > Green > Blue
Green > Red > Blue

If we are constrained by the preference of red over blue, then we see that of the three potential orderings of preferences, green is also preferred over blue two times out of three. So is choice rationalization going on, or is this merely an exercise in probability? Certainly it suggests that no conclusions can be drawn by a 66% chance of green trumping blue – this is what one would expect. One would hope that this is an isolated case, and the majority of cognitive dissonance theory is immune from Monty Hall’s crafty ways, but if the article is any indication, there isn’t entirely a consensus on whether or not this is the case.

*
Sometimes in mathematics, one encounters a problem whose solution is so counterintuitive that even when one completely understands the path to the solution, and is certain this path is correct, the final answer seems too strange to be true. One can simultaneously know the answer to be right, and still not believe it. For many people, the Monty Hall problem seems to fall into this category. Why does it conflict so much with our intuition, and how can we make peace with this seeming contradiction?

The answer to the first question is clear: we intuitively expect the probability of success if we switch to be 50%. There are two doors, one has a prize and one doesn’t, so there should be a 50/50 shot, right? And indeed, such a claim is supported by the statement that given doors 1, 2, and 3, the probability that the prize is behind door 1 given that it is not behind door 3 is 50%.2 So, what goes wrong?

What goes wrong is that, in asserting there is a 50/50 shot, we are forgetting the rules of the game. Monty Hall doesn’t show us one of the doors with a goat and then ask us to pick one of the remaining doors; we pick a door first, and based on our choice, Monty Hall picks a door to show us. Our odds increase to 66% because Monty can never reveal what’s behind the door we’ve chosen. This is important: this tells us that if we pick a door with a goat, Monty is forced to show us the other door with a goat – he has no choice in which door he can reveal.

Let’s imagine a variation on the problem: one in which Monty still reveals a goat, but he can reveal what’s behind the door we chose (provided we have not chosen the winning door). In the event that he shows us what’s behind our door, we are allowed to choose again between the two remaining doors. Assuming the hosts selects the doors at random, the odds are now in line with our expectations: there is a 50/50 chance of winning the car.

The reason why can be found in the following diagram. Here we assume that you will always switch, given the choice between staying or switching, and when you do have to choose again, you choose randomly. Also, as before, we assume that initially you choose door number 1 (the probabilities are the same if you initially choose door 2 or door 3).

You can click on the above image for a larger view.


Since 1/6 + 1/6 + 1/12 + 1/12 = 1/2, indeed we see that in this modified game, you gain no advantage from switching.

What do we conclude? The advantage in switching in the classical Monty Hall problem comes from the fact that our choice of door can potentially restrict which door Monty shows us. In this modified version of the game, Monty has no restriction other than the fact that he can’t show us the car. When the host is allowed to show you either one of the bogus doors, regardless of which door you pick, then the advantage to switching vanishes. Once this subtlety is understood, hopefully the seeming contradiction becomes untangled.

For discussion of some other versions of the Monty Hall problem, a good place to start is this Wikipedia article. And for those of you who are still unconvinced, this interactive simulation of the Monty Hall problem will, I hope, bring you to the light.

Make sure to join us next time!

1. Krauss, Stefen and Wang, X.T. (2003). “The Psychology of the Monty Hall Problem: Discovering Psychological Mechanisms for Solving a Tenacious Brain Teaser,” Journal of Experimental Psychology: General 132(1). Retrieved from http://en.wikipedia.org/wiki/Monty_Hall_problem.

2. Let A, B, and C be the events that the car is behind door 1, 2, and 3, respectively. We want to show what our intuition tells us: given the car isn’t behind door 3, the probability it’s behind door 1 is 50%. For those with some background in probability, this is a straightforward exercise in conditional probability. Using the given notation, we have

P(A | not C) = P(A and not C)/P(not C).

But P(A and not C) = P(A), since if the car is behind door 1, clearly it is behind door 1 and not behind door 3, and vice versa. Thus

P(A | not C) = P(A)/P(not C) = (1/3)/(2/3) = 1/2, as desired.

August 8, 2008 | 1 Comment »

Math in the News: Worms Love Calculus?

Those of you itching for some news last weekend may have noticed the following article, which was briefly featured on the front page of Yahoo News. In short, the article discusses the results of an experiment on the brains of roundworms. The experiment indicates that roundworms can mentally compute changes in salt levels with respect to their position in order to find food. Anyone who’s taken a bit of calculus may recognize that hidden in this is the notion of a derivative. In essence, concludes University of Oregon biologist Shawn Lockery, the worms use calculus to survive.

More computing power than an Apple IIe?

The notion that insects can do calculus is certainly good for a headline, and from a pedagogical standpoint it may be useful, although somewhat insulting to those who have trouble with math: “If worms can do calculus, anyone can!” All that aside though, isn’t the claim a bit disingenuous?

The idea that calculus is related to the ability of animals to find food should make sense, and indeed the article points out that it is believed a wide variety of species (including humans) do this. Think about it: if you are hungry, and you smell a barbecue, in order to find that delicious food you will most likely walk in the direction where the smell is the strongest. As the smell increases in strength, you will hone in on your direction – and if the smell strength decreases with your motion, it is likely that you will change direction, in order to zero in on the origin of the scent.

In essence, this is what the article says roundworms do in soil: they use salt levels to determine likely sources of food. Certainly, one can model this type of behavior using calculus, where the position of the worm (or the barbecue craving human) depends on the rate of change in the food’s scent at that position. The conclusion is that both species will move along the path of greatest marginal increase in scent.

But in what sense can it be said that the worms are “doing” calculus? Only in a very broad sense, it would seem. Many people who can smell out a great barbecue know nothing about calculus, so what does it mean to say that they are doing calculus in order to find their food?

Wouldn’t it be more appropriate to say that both behaviors can be modeled by calculus? Then the point of the article shouldn’t be that worms can do this kind of math, but that their behavior for finding food can be modeled in a relatively simple way using basic calculus.

This speaks, of course, to what many mathematicians will try and tell you: that math is simply a language in which to model the natural world. In a sense, then, calculus should be easy, because there are so many examples of it all around us. The problem is in formalizing these phenomena into an appropriate language, and in finding the best ways to teach this language once it is developed.

Hopefully this article can give mathematicians something else to point to when they assert that calculus really isn’t as scary as everyone makes it out to be. At the very least, when you are next invited to a dinner party, you can look smart by saying you used calculus to find your way.

August 3, 2008 | No Comments »