Math Goes Pop! » Animal Math

Let’s Make a Deal with Paul the Octopus

Matt — Wed, 14 Jul 2010 15:00:40 +0000

As summer reaches its midpoint, we come to the end of another rousing year of World Cup soccer. As with any international sporting event, fans all over the world have undoubtedly had their share of ups and downs. Of all the countries in this year’s tournament, however, I think Germany may be receiving the most attention, for even though they didn’t make it into the finals, the Germans have one thing no other country has: a precognitive octopus.

At least, that is what the media would have us believe. For the past several weeks, Paul the Octopus has captured the hearts, minds, and stomachs of people around the world. He’s a charming octopus, to be sure, but it isn’t his good looks that have gotten him this far. Instead, it’s his seeming ability to correctly predict the outcome of soccer matches. As time has gone on and Paul’s predictions have continued to prove themselves accurate, the amount of press he has received has only increased. Articles about him are everywhere on the internet: here‘s one discussing public outrage after he correctly predicted Spain to defeat Germany in the semifinals, and here‘s an article discussing his preference for Spain over the Netherlands in the finals. Search for “Paul the Octopus” in Google News and you will find thousands of results.

This is one popular mollusk.

Of course, I suppose it’s possible that Paul really can see into the future, at least as far as soccer is concerned. After all, he did correctly predict the winner of every game asked of him; an impressive feat, seeing as how his advice was requested a total of 8 times. However, Occam’s Razor suggests that we should look for a simpler explanation.

A natural first choice is to guess that Paul was simply guessing randomly, and is very lucky. The odds of this happening are small – assuming he has a 50% chance of picking correctly, the odds of him being right each one of the 8 times he predicted a winner in this World cup would be 1/2⁸, which is only approximately .39%. Very low odds indeed.

This analysis ignores biases that may be present – in particular, Paul’s octopus vision may bias him towards certain flag designs (which, given the fact that he frequently chooses Germany, seems plausible). The Wikipedia article I linked to discusses other sorts of possible bias. However, these biases would only influence which box he selected – they wouldn’t necessarily affect the odds that his selection would correspond to the winning team (although it is possible that he is being persuaded to throw his chips in for the favored team, which would increase the likelihood of his success). Either way, questions concerning how Paul makes his selection are more interesting to me, so let me focus for the moment on that.

First of all, one could easily argue that unlike flipping a coin, the trials here (i.e. Paul’s selections) are NOT independent. Indeed, what’s going on here may be very similar to an article in the New York Times a couple of years ago that discussed the lurking presence of the Monty Hall Problem in a classic experiment from psychology (which I discussed here).

The idea is quite simple. Paul chooses between two opponents in the World Cup by selecting a piece of food from one of two boxes. Each box is labeled with a country’s flag, and this is the most obvious distinction between the boxes. Suppose, for the sake of argument, that Paul has in his mind a ranking of his preferences for the flags, starting with the one he likes the most, and ending with the one he likes the least. Assuming between any pair of flags Paul prefers one to another, one could theoretically determine his preferences by giving him sufficiently many pairings of different flags. Moreover, assuming he does have preferences, the game selections are no longer independent, because each game gives us some information about his preferences.

Let’s dig deeper and look at his selections throughout the World Cup. Paul gave predictions for 8 games, and those 8 games involved 9 separate teams (only one game did not involve Germany). In the first game, Germany versus Australia, Paul selected Germany. Let us note that as: Now, already this gives us information about Paul’s preferences. There are 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 9! = 362,880 possible ways to order these 9 teams, since you can choose any one of the 9 teams to be your favorite, any one of the 8 remaining to be your second favorite, and so on. But given the above information, we already know that any choice with Australia ranked higher than Germany can not match Paul’s preferences. This eliminates a surprising number of possible outcomes – half, in fact, since for every list in which Germany is ranked higher than Australia, we can flip these two countries to obtain a ranking in which Australia is higher than Germany.

This then affects the probability of all subsequent pairings! Let’s take a look at the next game. In the second game, Paul correctly predicted Serbia to defeat Germany. We can write this as: This gives us even more information! Now, not only do we know that Paul prefers Germany to Australia, we also know he prefers Serbia to Germany.

Moreover, suppose (as seems reasonable) we assume the probability that he prefers Germany to Australia is 50%. Now, after the first match, we know he prefers Germany to Australia – the right followup question to ask is, what is the probability he prefers Serbia to Germany GIVEN that he prefers Germany to Australia? In particular, we have a conditional probability question here – these two facts are not independent! For example, the fact that Paul prefers Germany to Australia means that Germany cannot be last in his rankings – this should decrease the probability that he prefers Serbia to Germany. And indeed it does: supposing that each of the 362,880 rankings is equally likely, the probability that someone will prefer Serbia to Germany given that they prefer Germany to Australia is not 50%, but is only 33%!

To see why, consider an arbitrary ranking of the 9 teams. If we only consider the relative placement of Serbia (S), Germany (Ge), and Australia (A), there are 6 possible rankings among these three:

S > Ge > A
S > A > Ge
Ge > S > A
Ge > A > S
A > Ge > S
A > S > Ge.

However, if we also know that Ge > A, then rankings 2, 5, and 6 are eliminated, leaving us only with

S > Ge > A
Ge > S > A
Ge > A > S.

In particular, of these three remaining, Serbia is ranked higher than Germany only once. Therefore, the probability that S > G GIVEN that G > A is only 1/3, or 33%. (If you prefer, you can use a counting argument to show this as well.)

Of course, just as the first match gave us information, so did the second. Therefore, when it comes to the third match, we have even more information at our disposal. The third match was between Germany and Ghana, and Paul correctly identified Germany. In other words: Now the appropriate question to ask, of course, is: what is the probability that Paul prefers Germany to Ghana, given that he prefers Serbia to Germany and Germany to Australia? Well, we know that Germany can’t be his first or his last choice, because it must be preceded by Serbia and followed by Australia. Therefore, among these four countries, Germany must rank second or third.

If Germany ranks second, Serbia must be first, but we are free to make Australia third or fourth. Similarly, if Germany is third, then Australia must be last, and we are free to make Serbia first or second. In other words, we have four outcomes:

S > Ge > Gh > A
S > Ge > A > Gh
S > Gh > Ge > A
Gh > S > Ge > A.

Among these four choices, only 2 have Germany preferred over Ghana. Thus the conditional probability that one would prefer Germany to Ghana is again 50%.

The interested reader can easily continue on in this fashion. If you’re impatient, however, you can calculate the probability that Paul would have made the selections he did more directly. All we need to know is who Paul selected in each match. I’ll tell you that in the subsequent matches, Paul picked Germany over England, Germany over Argentina, Spain over Germany, Germany over Uruguay, and Spain over the Netherlands.

Given this information, suppose you want to know the probability that Paul selects Germany over Australia, Argentina, Uruguay, Ghana, and England, while he selected Spain over Germany and the Netherlands and Serbia over Germany. As stated before, there are 9! possible lists of preferences. In this case, it’s not hard to determine how many would lead to the behavior seen in this year’s World Cup. Since Paul picked Germany over 5 teams, but behind 2 teams, we know that Germany can only be ranked 3rd or 4th (any higher and there wouldn’t be room for Spain and Serbia above, any lower and there wouldn’t be room for the 5 teams below).

If Germany is ranked 3rd, then we can choose to put either Spain or Serbia in 1st place. Whichever one we don’t put in first place will then need to go in 2nd place, so that both countries are ranked higher than Germany. After that, we are free to order the remaining countries however we like. In other words, we see that there are 2 x 6! = 1,440 possible lists of preferences if Germany is ranked 3rd:

2 choices
1 choice
Germany
6 choices
5 choices
4 choices
3 choices
2 choices
1 choice.

Meanwhile, if Germany is in 4th place, we need to figure out how many ways there are to choose the three teams above it. Notice that since Australia, Argentina, Uruguay, Ghana and England must all be ranked lower than Germany, the three countries ranked above it must be Spain, Serbia, and the Netherlands. However, since we also need the Netherlands to be ranked below Spain, this only gives us three possibilities for the ranking of the first three teams: Spain, Serbia, Netherlands; Spain, Netherlands, Serbia; and Serbia, Spain, Netherlands. Once we have made that selection, however, we are free to choose the 5 teams below Germany however we please. In other words, if Germany is 4th, there are 3 x 5! = 360 possible lists of preferences.

Combining these, we see that there are 1,800 possible lists of preferences that would lead to the behavior shown by Paul. Since the total number of outcomes is 9!, this gives a probability of only 1,800/9!, or roughly .49%. This is the same value you will get if you calculate the remaining conditional probabilities and multiply them together.

Of course, if one wants a more impressive number, one can always try to correct for bias in Paul’s selection. For example, suppose we assume that Paul will never choose England or Australia if given the option of a different flag – this seems reasonable, given experiments on how his species sees (the other flags have more contrast and are more focused on horizontal shapes, which apparently his species is drawn to). If we make this assumption, the number of potential preference lists drops to 7! x 2 = 10,080, in which case the probability that Paul would choose as he did jumps up to 17.9%!

There are valid concerns about this model, though. For instance, given the choice between two flags, why should we assume that Paul will always choose the same one over the other? Equivalently, why should we believe that Paul’s decisions follow a prescribed preference list? Indeed, when Paul was used to make predictions in 2008, he selected Germany over Spain, unlike his selection of Spain over Germany in 2010. In 2008, it was Germany who was the victor, and so Paul guessed incorrectly – perhaps he learns from his mistakes, after all.

Whatever the case, I doubt this octopus has any special ability. And even if he does, I don’t know that that would necessarily be a good thing. For we all know that when 8 limbs are combined with super powers, nothing good can come of it.

Is this the future that Paul's powers portend? I believe it is.

Deep Sea Math Hunting

Matt — Wed, 23 Jun 2010 21:58:06 +0000

Every now and then an article pops up which highlights a link between mathematics and the animal kingdom, and I’ve been able to discuss several such links on this blog. The latest entry into this category concerns the movement of sharks (and other ocean creatures) as they hunt for food. A recent article in Nature has spawned a great deal of interest, and the topic has been discussed on the websites of Wired, Discovery, and Physics World.

What does the motion of sharks have to do with mathematics? Well, suppose you are a shark. Unfortunately, there are not yet any In-N-Out’s under water, so when it comes to food you are on your own. What would be the best way to forage for your food? With your heightened senses, you would undoubtedly be a formidable opponent in an area rich with prey, but what if you are in a more sparsely populated area? What’s the best way for you to search for nutrition? As it turns out, the best thing for you to do may be to follow a type of random motion known as a Lévy flight.

The following is from the aforementioned Wired article: “Computer models suggest Lévy flight is the optimal search pattern for predators in low-prey areas, and maximizes the chance of a random encounter.” Visually speaking, a Lévy flight is characterized by short movements in random directions, interspersed with occasional longer trips in a particular direction. Here is a sample from the Wikipedia article on the subject:

I bet little kids are awesome at drawing Lévy flights.

Suggestions of Lévy flight patterns in animals goes back to at least 1996, but a recent study led by David Sims of the Marine Biological Association Laboratory is the first to reach these conclusions on the basis of such a vast amount of data. Indeed, by tagging marine animals with GPS locators, they were able to track the movement of 14 different predator species over a period of 5,700 days! With some rare exceptions (such as the great white shark), they found that animals in areas with less food were more likely to move in a Lévy flight pattern.

Several articles (jokingly, I assume) point out that this should make us even warier of sharks, since they can use math to optimize their behavior. As is often the case with these articles, the headline rarely reflects the reality. Whether this behavior is learned or evolved, the fact that sharks may follow this pattern says as much about their mathematical ability as the resemblance of trees to fractal patterns says about their mathematical ability. Mathematics is a language that we can use to describe nature, so the fact that we can use mathematics to describe this natural phenomenon shouldn’t necessarily be that surprising. If a shark gets a 5 on the calculus AP, that’s when I’ll be surprised.

Here are some sharks in a group study session (courtesy of the Discovery article mentioned above).

Baby Animals Just Want to Do Math

Matt — Fri, 22 May 2009 22:58:00 +0000

Over the past few months there have been several studies aimed at understanding the mathematical sophistication of some of our friends in the animal kingdom. This is a topic I have discussed before, but these new findings are interesting and worth mentioning.

The most recent experiment involves the cutest animal discussed so far: baby chicks. Don’t let their looks fool you, my friend, for under that puff of yellow down sits a mind capable of mathematical wizardry. Surprisingly, researchers found that chicks were not only able to perform simple mental calculations, but could do so from a very young age.

How do you tell if a baby chick can do math? Well, apparently the little ones try to stay close to familiar objects (for example, their mother). Moreover, given the choice between a small group of familiar objects and a larger group of familiar objects, researchers noted that chicks tended to gravitate towards the larger group.

But what if some calculation is required to determine which is the larger group? Researchers put the chicks in a glass cage and then hid yellow balls behind one of two screens. Sometimes they would then transfer some balls from one screen to another, in a process that the chick could see. However, the chick couldn’t see how many balls were behind each screen, so the only way to keep track would be to keep track of how many balls moved from one side to another, and how many were initially on each side – in essence, to perform some basic mental arithmetic.

Surprisingly, the chicks were up to the challenge, and consistently went towards the larger group, even though the two groups were hidden from view. Here’s a link to a video that shows the basics of the experiment design.

Apparently, chicks love yellow plastic balls.

Of course, the word “baby” has several meanings. Baby chicks are described in this way because they are young, but the adjective baby could just as well describe tiny things (think of baby corn, baby back ribs, or baby math blog readership). With this interpretation, baby chicks aren’t the only baby animals that want to do math – some baby fish are joining the party as well.

This is a combination Math/French joke.

Please meet the graceful mosquitofish, a species poised to revolutionize mathematics as we know it. Or, if not that, at least it can do some simple counting, according to researchers from the University of Padova in Italy.

What makes us think these fish can count? Well, the fish were put in a tank and given the choice of several doors to swim through. One of those doors had a larger group of mosquitofish (no doubt they were all studying for the Putnam exam together). First the researchers trained the fish to associate the correct door with a certain number of geometric shapes. The fish were then put in an empty tank and were allowed to move freely through any of the doors.

The results? More often than would be expected by chance, the fish chose the door with the number of shapes that they had been trained to enter. Moreover, to try and pin down the effect of the number of shapes, rather than any other parameter, researchers “placed sets of shapes that varied in size, brightness, and distance…only the number of shapes stayed the same.”

Does this mean that these tiny fish have some rudimentary method of counting small sets? Do they have a number sense? What does it even mean to claim that a fish can count? With further research, maybe the answers to some of these questions will become clear.

Our last foray into mathematics within the animal kingdom comes to us from what is undoubtedly the coolest looking animal mentioned so far: the rhesus macaque.

Researchers at Duke University were able to have “widespread success” in getting rhesus macaques to calculate differences of whole numbers.

The main idea is similar to what was done with the chicks, although slightly more was expected from the macaques: they were first shown a collection of dots on a computer screen. The dots were then covered by a square, and some of the dots flew off screen – the monkey could see how many dots were removed, but not how many dots were remaining. The article linked above has a video showing this animation.

Afterward, the monkeys were given a choice between two collections of dots – one with the correct number of dots remaining, and one with the incorrect number of dots remaining, and were asked to pick a collection. Researchers found that the macaques performed just as well at identifying the correct difference as the human college students that were used as a control. (Then again, the macaques were rewarded for their correct answers with Kool-Aid – no such incentive is mentioned for the human controls.)

Could the secret to mathematical ability be locked inside the belly of this anthropomorphic glass pitcher? The question remains open.

With all of these stories, there is an important question to ask: why should we care? Who cares if chicks can count, or if macaques can subtract dots? More generally, why should we be bothered with questions regarding the mathematical ability of other species?

One important answer is that clues about the abilities of other species may help give us clues as to how our own ability to do math has evolved. More specifically, we can attempt to address the question: what is the role that evolution has played in the development of mathematical ability?

A few of the articles mention potential evolutionary benefits to mathematical ability. For example, in the case of the mosquitofish,

…the ability [to count] in fish is probably a “last resort” strategy that has evolutionary underpinnings, [lead study author Christian] Agrillo said.

That’s because non-numerical cues probably come more easily to fish as they make rapid-fire decisions.

Being able to count may require more brainpower than simply judging numbers based on size. But counting might sometimes be necessary as the fish seek safety in numbers to shield themselves from predators, Agrillo said.

This “safety in numbers” phenomenon may also help explain the chicks ability to keep track of small sets of numbers. If there is an evolutionary advantage to moving towards a larger group, then it’s reasonable to guess that chicks may have developed a basic ability to keep track of relative sizes, even under difficult conditions such as the ones present in the study.

What about the macaques? In this case, there may also be an evolutionary advantage to having a knack for mathematics. The authors note that “For instance, research has shown that apes can determine at a glance roughly how much food is present in an area and decide whether to stay and eat or to move on.” This ability to estimate would require at least a certain level of mathematical sophistication, one which could arguably depend upon the ability to perform simple subtraction calculations.

So, there are evolutionary arguments for the development of mathematics – but to what extent it can be said that these animals are “doing math” is a good question. And as for how to bridge the gap between their level of mathematical sophistication and abstract thought and ours, I’ve no doubt there is plenty of research waiting to be done.

I would start by looking into the Kool-Aid.

Math Gets Around: The Entomology of Civil Engineering

Matt — Thu, 12 Feb 2009 02:40:00 +0000

In the continuing saga of animals that are better than you at math, it now appears that ants are much better than most of us at optimization. Granted, they may not be able to think abstractly, but in concrete terms, they far surpass us with a particular type of optimization: the efficiency of traffic flow.

As anyone who has gone to a picnic will tell you, ants do a very good job of creating traffic streams – their foot traffic moves steadily, and without the major pileups to which my fellow residents of Los Angeles have become so accustomed. One could argue that the wide expanse of park area is proportionately much larger for the humble ant than what most motorists have to live with, but even so, the march of the ant colony often appears quite regimented, even with space enough to make a wider path. How is it that ants can control their traffic so well?

This article from the Wired Science blog discuss how ants succeed where we fail. At the heart of the matter is a study from the University of Sydney on leafcutter ants. In order to give the ants a better sense of what it’s like trying to navigate through a congested urban landscape, scientists restricted the ants to naturally narrow pathways, such as the ends of tree branches, in order to better understand how these ants organize their traffic in cramped spaces.

With their superior understanding of traffic flows, could ants one day dominate the world? Some scientists say “Yes!”

The findings clash with most people’s behavior on the freeway:

In the latest findings, published in the February issue of the Journal of Experimental Biology, [entomologist Audrey] Dussutour’s team found that ants leaving the colony automatically gave right-of-way to those returning with food. Of the returning ants, some were empty-mandibled — but rather than passing their leaf-carrying, slow-moving brethren, they gathered in clusters and moved behind them.

Rather than try to outpace their slower moving brethren, those without loads to carry simply kept pace with the slower ants. This is at direct odds with what most people do on the roads – who wants to drive stuck behind a bus? Based on our own behavior, we may question the wisdom of the leaf cutter ant’s process.

As is often the case, however, nature knows best. By not trying to barrel ahead of the slower moving ants, the ants without any baggage saved time on average. Not by a paltry amount, either – the study estimates “that patience reduced the average delay experienced by an individual ant crossing a crowded three-meter bridge from 64 to 32 seconds.” That’s a 50% reduction in commute time!

One plausible explanation for the difference between our behavior and the ant’s behavior is that we are looking at different optimization problems. People in general are trying to minimize their own individual travel times, and the other cars on the road aren’t given much consideration. With (apparently) smaller egos, the problem in the ant’s case is to make the whole traffic network run as smoothly as possible, so food can be brought in quickly, and energy isn’t wasted in traffic jams.

The study helps give weight to the maxim that patience is a virtue. Haste while driving carries with it certain risks, risks that on average far outweigh the benefits that come from not trying to outpace others on the road.

It’s doubtful that this study will do much to change human behavior, but understanding efficient traffic flow algorithms certainly has its applications, from urban planning to the engineering of self-driving cars. Perhaps people would be more patient if they weren’t the ones doing the driving.

Unfortunately, the days of the self driving car are not yet upon us, so until that day arrives, we must be content with what we have. So, dear reader, take a cue from the noble ant, and slow it down when you’re on the road – over time, it may save you time.

A glimpse into cities of the future?

Math in the News: Elephants are Smarter than your Babies

Matt — Fri, 21 Nov 2008 02:31:00 +0000

I missed the memo on this one, but apparently worms aren’t the only animals capable of doing math. A recent experiment coming out of the University of Tokyo suggests that Asian elephants have an unexpected aptitude for arithmetic. While many animals have a rudimentary counting ability, and are able to distinguish between sets with only a few elements, it seems that elephants are able to take things a step further, and can consistently differentiate between larger numbers such as 5 and 6.

Is this difference significant? Within the animal kingdom, it would seem so. Here’s how it breaks down, courtesy of this article:

A theory held by some is that humans and other animals share a basic neural system called an “accumulator” that can clearly distinguish numbers of objects less than three or four but that cannot reliably discriminate between bigger numbers. This accumulator is active in animals and, perhaps, in human infants, the theory contends. Higher-order number abilities require the collaboration of other, more highly developed brain systems found only in humans.

An ability to consistently distinguish between larger number (by larger, I mean larger than four) may therefore indicate a more advanced accumulator system than is found among the general kingdom’s populace.

What does this mean for the noble elephant? While it’s certainly a bit premature to start hiring them as our accountants or financial advisers (although, given the current economic conditions, perhaps giving elephants access to our finances isn’t such a bad idea), it certainly does highlight what those active in elephant research already know: these majestic creatures aren’t all looks. Each one has a head on its shoulders as well.

This elephant is no doubt pondering some very deep mathematics. The picture is taken from a collection that accompanies an excellent National Geographic article on these mathematical savants.

It is natural to ask what sort of evolutionary process would lead to the elephant’s surprising counting aptitude (aside from the obvious benefit of being able to impress the ladies). An article from the London Times suggests the following alternative hypothesis:

Speculation among scientists over why the elephant should have developed its limited but nonetheless impressive mathematical ability centres on the way in which the lumbering creatures move in herds. A basic counting ability, say experts, might act as a guarantee that no calf is left behind.

Is the acquisition of mathematics knowledge driven by evolution? Perhaps in the animal kingdom, although if you ask graduate students in mathematics, I doubt they will say that an aptitude in math has really helped them to propagate the species. Those days are coming, my friends, but they are not here yet. For now, let us find solace in the fact that when it comes to defending the belief that mathematics is of fundamental importance, we will have a mighty ally in the Asian elephant.

The results of the experiment came as no surprise to Babar, whose sharp intellect not only allowed him to become king of the elephant empire, but also blessed him with a keen eye for fashion.

Math in the News: Worms Love Calculus?

Matt — Mon, 04 Aug 2008 00:46:00 +0000

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.