Ever since I saw this awful page about exponential growth (namely zebu.uoregon.edu/1999/es202/l3.html, which I complained about earlier), I've been considering the question of what a good piece of curriculum would look like for teaching exponential growth.
With apologies for diving right in, let me share a few working hypotheses:
1. Fluency with the mathematics of the exponential function does not automatically or inevitably lead to a rich intuitive grasp of exponential growth.
I arrived at this hypothesis by reflecting on my own educational trajectory. Though I mastered this body of mathematics as a teenager, I would say that my instinctive feel for exponential growth has become strong only in the past few years. (In fact, I wonder if my technical facility with mathematics actually shielded me from ever having to develop a rich mental idea of exponential growth.)
2. Fluency with the mathematics of the exponential function is not even necessary for having a rich intuitive grasp of exponential growth.
This is stating the case strongly; but for now I'm interested in pushing this perspective as far as I can.
3. A great piece of curriculum for exponential growth would be a valuable, eye-opening, and even transformative experience for a wide variety of audiences, including college students of all kinds, college faculty members, and adults outside of academia.
I don't have this magic piece of curriculum yet, but what I assembled recently for my Rediscovering Math class was perhaps a small start. A portion of what we covered is reproduced below. A leisurely ramble it may be; but I would also say that here & there it contains some real mathematical insights about exponential growth. (With respect to Hypothesis #1 above, I should say that I arrived at some of these insights for the first time as I prepared to teach this class!)
We begin, as one might expect, with vampires.
A biologist on the Bennington faculty pointed me to this amusing paper by two physicists that aims to debunk various items of folklore about ghosts, vampires, and zombies. The vampire section was one of those surprising examples of exponential growth. The authors point out that according to standard (pre-Anne Rice) vampire lore, vampire-human ecology is simply a non-starter. The authors argue that with vampires feeding on people who turn into vampires who feed on people who turn into vampires who feed...and so on...then it would only take about three years for the entire world's food supply to run out!
When I saw this argument, my first reaction was embarassment that the absurdity of vampire population dynamics has always been right in front of my face without my ever having noticed it. My second reaction was to defensively poke holes in the argument. For example, the authors conclude by reductio ad absurdum that there's no such thing as vampires (or else we'd all be vampires by now); but we might alternatively conclude from the reductio that we're all just about to be vampires, or that vampires must have natural enemies, such as werewolves. (As in the "Blade" movies. Perhaps Hollywood understands exponential growth better than most.)
I encourage you to read the article to get a sense of how the numbers work out. But for the sake of time, let's move on.
Paradoxes of paper folding
There's an old saying that you can't fold a piece of paper more than seven times. In class, we tried it with 8.5x11 sheets of paper, and everybody managed the same number of folds—just six. Then we went out in the hallway and tried another folding experiment, this time folding a very long sheet of paper towels, over a hundred feet long. (We only used lengthwise folds in this case.) As it turned out, the difference between a single sheet of paper and a hundred-foot-long strip of paper was only a single fold! Seven, instead of six.
It was fascinating to enact the folding process for the long strip. After the first two or three folds, everything seemed to be going fine. Then, when we went from fold #5 to fold #6, the game was suddenly up. (More about the suddenness of exponential growth below.)
But why is paper folding an example of exponential growth at all? There's clearly some sort of doubling going on—or, what is the same, some sort of halving. And somehow this must be related to the difficulty of persisting in the folding process beyond a very few steps. But to draw the connection more clearly, I presented the students with a rough mathematical model of paper folding. The derivation is shown pictorially below; it leads to the equation L/t = 2^(2N), where L is the length of the strip of paper, t is the thickness of the paper, and N is the maximum number of folds obtainable. Though the model is crude, it does reveal the exponential nature of the process, and it shows N to be a function of the length-to-thickness ratio, as we would expect. Using this formula, we were able to estimate the number of times one would be able to fold a strip of paper that initially encircles the earth along the equator. (Guess how many!)
There's a nice echo of this in Philip and Phylis Morrison's excellent book The Ring of Truth, based on their 1987 PBS series. The Morrisons explain that expert Chinese chefs are able to make incredibly thin noodles called "Dragon's Beard" by repeatedly stretching the noodles by hand and cutting them in half at each step. In two minutes, the chef interviewed in the Morrisons' book has achieved twelve doublings, yielding about four miles of noodles, each noodle about twice the thickness of a human hair. Legendary chefs of the past were said to attain thirteen doublings.
Grains of rice
We all know the paradox of the Chinese emperor and the grains of rice. The way I tell the story, a wise man does a favor for the emperor, and the emperor asks what he might do in return. The wise man asks for 1 grain of rice to be placed on the first square of a chess board on the first day, 2 grains to be placed on the second square on the second day, 4 grains to placed on the third square on the third day, and so on, doubling the number of grains each day. The emperor agrees, and after a couple of weeks, all of the rice in the empire belongs to the wise man!
The night before class, I wondered whether I might attain a greater understanding of this paradox by acting out the wise man's challenge. So I sat on my kitchen floor and arranged 1+2+4+8+16+32+64+128+256 = 511 objects into a geometric pattern.
(The largest objects are dried beans, then it's rice grains, and then anise seeds.)
Just as in the experience of making "A Few Iron Posts of Obseration", I found it instructive to "think with my hands" for a while. I sat peacefully on my kitchen floor, pushing the pieces around, planning the next stage, and repairing damage from the occasional errant finger or too-vigorous exhaling of breath. With my hands busy, my mind was free to roam. I reflected on the way my ever-shrinking materials —beans, rice, seeds—resembled the ever-shrinking computer chips that carry out our society's calculations. If I needed to take that next step to an outer tier of 512 objects, how would I fit them into the structure? What objects could I use? Salt grains? How then would I manipulate such tiny objects and put them in the proper places? How would I better control my breathing and other destructive effects?
Likewise, how will we continue to shrink our processors to reach the next tier? What will we make them out of, and how will we assemble their circuits? How will we protect them from environmental interference? Can our ingenuity keep up with Moore's Law forever?
The latest model I've come up with for explaining exponential growth is something I call "the Megamountain." Here's how it goes.
We're going to imagine climbing a mountain. First, hink about what it would be like if the mountain had the same steepness all the way up. What would this mountain look like? Try to draw it.
Next, think about what it would be like if the steepness of the hill kept increasing steadily as you went up. What would this mountain look like?
These first two mountains look something like the cartoons shown below.
Now for the mountain you don't want to climb: The Megamountain. On the Megamountain, the steepness of the mountain at any point is proportional to the altitude at that point.
That's the rule of the Megamountain: The steepness is proportional to the altitude.
If you think about this rule carefully, then you begin to realize that the Megamountain is a runaway situation. Because if you're high up, then [by the rule] it's steep; but, if it's steep, then because of that your next step gains a lot of altitude; but [by the rule] that means it's now going to be even steeper; but that means your next step will gain altitude even faster than before; but that means it'll now be even steeper; and...AAAHHH! It makes my head hurt to think about it!
When I think about what it would be like to climb the Megamountain, I actually get a panicky feeling that I can't possibly keep on going this way. I don't even want to take that next step, because every step is feeding a vicious cycle.
(By the way, you'll notice that I'm not going to try to draw the Megamountain. That's because it can't be drawn; not really. Sure, you can plot a graph of y = e^x, but in the end, you're going to find yourself plotting only the region around x-values of order unity. And in this region, the curve looks roughly similar to a parabola, so you haven't shown what is special, and terrifying, about runaway exponential growth.)
A collection of runaway situations
There are a lot of runaway situations like the Megamountain, including:
* Unchecked population growth: The number of babies born is proportional to the number of people already here. More babies make more people make more babies make more people....
* Gestation: Suppose you had to build a hundred billion houses in 9 months. I think you would quickly hit upon the idea of building houses that build houses. This is how we get from a single fertilized egg cell to a big fat baby in only 9 months. The cell is a house that builds houses. Like grains of rice on a chess board, the number of cells added is twice the number of cells that were there before. More cells make more cells make more cells make more cells....
* Chain reactions. A uranium nucleus splits into two, and the two products strike two more uranium nuclei, causing them both to split in two; their four products strike four more uranium nuclei, and so on. This process is called fission; it was actually named for the biological process of cell division (which was called fission first). In the same way that a big, chubby baby begins with a single cell, here a single subatomic "pop" is magnified, in a millisecond, into an explosion that can level a whole city.
Mathematically, a pregnancy is a runaway chain reaction in the uterus...an explosion of a kind, but one that takes 9 months to unfold.
* Compound interest. The amount of money credited to your account is proportional to the amount of money already there. More money makes more money makes more money makes more money....
But you know, I have had savings accounts, and I have never exactly had the feeling that my money was undergoing an explosive chain reaction! The reason is that the interest rate is so small. It's true that if you wait long enough, your money will double, then double again, and eventually bankrupt the Chinese empire. But the question is, how long will it take to double?
The rule of 72: Divide 72 by the interest rate, and that's how many years it will take to double.
Example: You have a CD earning 4 percent interest. Divide 72/4 = 18, so your money will take 18 years to double. After 18 more years, it will double again.
Warning: The cost of goods and services is also growing exponentially at a 4 percent rate (at least), so by doubling your money in 18 years you are really just keeping up. Your $100 today will double to $200, but that $200 will only buy what $100 buys today. Hence, if your money is not earning at least the same as inflation, the growing cost of goods and services will outstrip the value of your money, and you will actually be losing money in real terms. This is called "inflation risk," and it's the reason you have to put at least some of your money into higher-risk/higher-return investments.
The suddenness of exponential growth
I like to show people these two crude movies that I made a long time ago. Both movies are cartoon visions of what it might be like to ride in a spaceship that splashes down on the north pole. In the first movie, the spaceship moves at a constant speed. In the second movie, the spaceship moves at an exponentially increasing speed.
(The views are through a porthole on the spaceship. Sorry about the aspect ratio - something got screwed up when I put the videos on YouTube.)
* Note, a clearer version of the first movie is available here.
* Note, clearer version of the second movie is available here.
Whenever I show people these videos, they can hardly believe the second video. It says a thousand words about the way exponential growth can sneak up on you—and how unstoppable it is, once it gains momentum.
Where does the suddenness of exponential growth come from mathematically? One way to think about it is to recognize that when we "run the clock in reverse," exponential growth is a continual process of cutting in half. This means that any process of exponential growth must spend a very long time at very small values. And in any graphical or visual sense, the point is that one very small number is going to look visually just like another, even if the two numbers in question differ by many orders of magnitude. (On a graph with values ranging from 0 to 1, a value of 0.0003 is going to be indistinguishable from a value of 0.0000000008—even though you'd much rather your chance of winning the Lotto were 0.0003 instead of 0.0000000008!)
Additionally, when you throw in the fact that the rate of change of any changing quantity involves taking simple differences, you see why the rate of change can remain small even when the underlying numbers are actually growing by orders of magnitude. (The difference between two small numbers is necessarily small, even when the two numbers differ by orders of magnitude.)
All of this is why you can watch something "growing exponentially in time" and wonder why it's just sitting there. It's just sitting there, sitting there, sitting there, and BANG! All of a sudden it explodes. The explosion happens when your numbers take the crucial steps from "smaller" to "small" to order-unity. Prior to order-unity, it looks like nothing is happening; after order-unity, it's too late to do anything.
Ultimately, "understanding exponential growth" might have little to do with being able to solve certain classes of transcendental equations. It might instead depend on having a gnawing feeling in your gut that exponential growth is an unstoppable force: not only an unclimbable mountain, but an insatiable feeding machine that will devour anything in its path; and a monster that will lie in wait, lie in wait, lie in wait, and then leap forward in the blink of an eye.
But to end on a more positive note, we should also remember that exponential growth can also be a resource. (Anybody need four miles of noodles? Just fold 12 times!) In other words, the best weapon we have against the exponential function might be the exponential function itself. This idea arises for example in the theory of quantum computing, which, when it gets here, will be a process in which the computational resources at our disposal grow exponentially with the number of particles in the processor. Previously intractable problems will become solvable in an instant!
Exponential growth as a resource also comes up in tipping point phenomena - you tell two friends, and they tell two friends, and so on and so on and so on. The numbers stay small, but they're working their way up the orders of magnitude, until we reach the fateful stage of order-unity. We usually think of this model in connection with epidemics and fads. But what if the thing we're spreading is instead a message of positive social change: such as one about changing our habits of energy consumption? That would be fighting fire with fire.