SOME EARLY ESTIMATES, SPECULATIONS

Postponing discussion of more engaging biographical issues and more intriguing mathematics, I'll begin simply with my early enchantment with numbers, the atoms of the mathematical world. Long before the Count on Sesame Street, I loved to count. I counted anything and everything, including, my father told me, the number of little thin cylinders in a pack of his ubiquitous cigarettes, although I doubt he appreciated the cigarettes being all grubbed up with my sticky toddler fingers. I mentioned in the introduction my numerical issues with Santa Claus. A considerate little fellow, I also remember humoring my parents whenever they mentioned him. I wanted to protect them from my guilty knowledge of his nonexistence, and so I feigned belief. (This is a nice illustration of the difference between the mutual knowledge my parents and I had—of his nonexistence in this case—and the common knowledge we didn't have, not only knowing this fact but also knowing that the others knew, knowing that the others knew that we each knew, and so on.) My brother Paul, three and a half years my junior, was only a baby, so it wasn't him I was trying not to disillusion.

In any case, my qualitative “calculations” had proved to me that there were too many expectant kids around the world for Santa Claus to even come close to making his Christmas Eve rounds in time, even if he didn't stop for hot chocolate or bathroom visits, a pressing concern for me. This may sound like quite a pat memory for an author of a book titled Innumeracy to have,1 but I do remember making rough “order-of-magnitude” estimates that showed Santa to be way overextended.

An oddly vivid memory of a similar idea dates from my days as a fifth grader. (It's a tiny bit jarring to even record that I had “days as a fifth grader.”) The teacher was discussing some war, and a girl in the front row asked how the losing country survived if all its people died in the war. She cried when she spoke, and it was clear she believed that every single soldier as well as almost all other people in a losing country died. I remember wondering if, common sense aside, she'd ever seen the aftermath to World War II depicted in any of the countless cheesy war movies I used to watch every Saturday afternoon. I was very shy in elementary school and only quietly came to my smug judgment about her numerical naïveté. Besides I really liked her and, in a ten-year-old sort of way, thought her emotionality cute. (I plead guilty to having been a ten-year-old sexist.)

Before resuming the narrative, let me flash-forward for the next few paragraphs to note that these particular memories are not mere personal remembrances, but rather they connect up with later concerns of mine about numerical estimation and innumeracy. The sad not so cute reality is that most adults don't have a much keener appreciation for magnitudes than my fifth-grade classmate. To cite a once popular American TV show, I think it's fair to conclude that they are in this sense not smarter than a fifth grader, especially when the issue is numerical estimation.

Some easy examples from a course in quantitative literacy that I often teach illustrate the point as well as they do the persistence of my fifth-grade judgmental attitude. For example, if I say to students in my class or to adult acquaintances that I heard that a Rose Bowl quarterback once shook hands with almost everyone in the stands after a startling come-from-behind victory, they are rightfully dubious that this would ever happen. Still, very few will note that, like Santa's journey, it is next to numerically impossible. Even if only half of the 100,000 fans or so at a game came down to shake hands and each handshake took, say, 4 seconds, that's 15 per minute, and 50,000 divided by 15 per minute is more than 3,300 minutes or about 7 8-hour days of handshaking. That would likely mean that the quarterback would never throw another pass.

Continuing with this perhaps jejune line of thought, I often get similar off-the-mark responses from students as well as educated neighbors if I claim to have read a headline claiming that experts fear housing costs (the total of rents and mortgages) will top $3 billion next year in the United States. They might respond by referring to the mortgage crisis, to greedy bankers, and the like, but they seldom will point out that the number is absurdly small—equaling about $10 annually per person for housing. And near total bafflement is the response to playful questions like “How fast does a human grow in miles per hour?” Alas, 62.38172548 percent of the time students and others take numbers as merely providing decoration, but not really imparting information.

Incidentally, a one-question diagnostic test for innumeracy is to ask someone to give very quickly and without resorting to a calculator the approximate average of these three numbers: 11 billion, 6 trillion, and 117 million. Only the numerate will answer 2 trillion. And a classic educational gambit assumes that the 4.5-billion-year history of the earth has been shrunk down to one year and asks you to determine how long before the end of the year various events occurred, say, the appearance of the most ancient religions as well as the appearance of you. Taking the first appearance of ancient religions to be very roughly 4,500 years ago and the first appearance of little old you, let's say, 45 years ago, the answers are only 30 seconds and .3 seconds, respectively, before midnight on December 31.

These little calculations are easily dismissible, but—and this is the point—I think they're examples of the same numerical obtuseness that afflicts many who think that a disproportionate share of American wealth goes to foreign aid, or that government earmarks for relatively paltry 50-million-dollar projects are the cause of the deficit, or that terrorism or Ebola, but not global warming, is a serious risk. And very few are aware that the two-trillion-dollar cost of the Iraq War is about 250 times the annual budgets of the National Science Foundation and the Environmental Protection Agency. Not unrelated is the mundane risk blindness of bicyclists I see driving down narrow, busy streets near my home in Center City, Philadelphia. Many don't wear a helmet, ride with no hands on the handlebars, text away on a cell phone, and listen to music through their earbuds. At the same time they might be very concerned about pesticide residue on the apple they are eating. Quite a stretch, I know, from beliefs about Santa Claus and bicyclists to foreign aid, war, and global warming, but this book contains at least as many general musings as personal remembrances, most of them, I hope, at least somewhat related.

To continue my story, at around this same time, the fifth grade or so, I began reading newspapers, an engaging introduction to which was provided by the Milwaukee Journal's “Green Sheet.” This four-page daily section was printed on green newsprint and was full of features that fascinated me. At the top was a saying by “Phil Osopher” that always contained some wonderfully puerile pun, a verbal category to which I'm still partial. There was also the “Ask Andy” column: science questions and tantalizingly brief answers. Phil and Andy became friends of mine. Were there a Twitter at the time and they had accounts, I would have been an avid follower and retweeter. And then there was an advice column by a woman with the unlikely name of Ione Quinby Griggs, who gave no-nonsense Midwestern counsel with which I often silently disagreed. Of course, I also read the sports pages and occasionally even checked the first section to see what was happening in the larger world.

Stimulated perhaps by Phil and Andy, I was captured at this young age by the idea of a kind of atomic materialism. I'd read that everything was composed of atoms, and I knew that atoms couldn't think, and so I “thought” this proved that humans couldn't think either. I was so pleased with this groundbreaking Epicurean idea (despite Phil, I didn't know the word yet) that I wrote it neatly on a piece of paper, folded it carefully, put it inside a small metal box, taped it securely, and buried it near the swing in our backyard where future generations of unthinking humans could appreciate my deep thoughts on this matter. I also remember speculating that just maybe there was a kid somewhere—in Russia, perhaps—who was as smart as I was. Pursuant to this I would scrawl “John is great” in secret places from closets to the attic, demonstrating either delusions of grandeur or simply youthful arrogance.

In any case the notion of emergent qualities, properties, and abilities didn't complicate my youthful certainty about these matters, and the dreary conclusion I came to that we couldn't really think was one I oddly found quite cheering. What I didn't find cheering was a recurring thought that some great new scientific discovery or philosophical insight would be announced, and I would find myself “seven brain cells short” of understanding it. It would be just beyond my personal complexity horizon. As a result of that irrational fear and of having read that alcohol kills brain cells, I resolved to be a lifelong teetotaler. My understanding of brains and conceptual breakthroughs has grown a little more sophisticated over the years, but the (almost) total teetotaling has persisted.

PEDAGOGY, VANQUISHING BLOWHARDS AND OPPONENTS, AND MONOPOLY

A bit later in elementary school I developed a very personal appreciation of mathematical certainty (as opposed to other sorts) that was germane to an adult concern of mine: mathematical pedagogy. Discussions of pedagogy and curricula too often tacitly assume there is a best way of imparting mathematical knowledge, igniting mathematical curiosity, and developing an appreciation for mathematics. There isn't. People's backgrounds, interests, and inclinations vary enormously and so should pedagogical techniques. As I've described in Innumeracy and mentioned in the introduction here, my interest in mathematics proper was partially a consequence of an intense dislike for my elementary school mathematics teacher, whose real occupation, it seemed, was being a bullying martinet.

I was very interested in baseball as a kid. I loved playing the game and aspired to be a major-league shortstop. (My father played in college and professionally in the minor leagues.) Even now I vividly recall the highlights of my boyhood baseball career: a game-winning home run over a frenemy's backyard fence and a diving knee-skinning catch in center field on an asphalt playground. My two worse moments: being beaned at bat by the local fastball ace and racing in from center field only to miss catching the ball that sailed over my head.

I also became obsessed with baseball statistics and noted when I was ten or so that a relief pitcher for the Milwaukee Braves had an earned run average (ERA) of 135. (The arithmetic details are less important than the psychology of the story, but as I remember them, he had allowed 5 runs to score and had retired only one batter. Retiring one batter is equivalent to pitching 1/3 of an inning, 1/27 of a complete 9-inning game, and allowing 5 runs in 1/27 of a game translates into an ERA of 5/(1/27) or 135.)

Impressed by this extraordinarily bad ERA, I mentioned it diffidently to my teacher during a class discussion of sports. He looked pained and annoyed and sarcastically asked me to explain the fact to my class. Being quite shy, I did so with a quavering voice, a shaking hand, and a reddened face. (A strikeout in self-confidence.) When I finished, he almost bellowed that I was confused and wrong and that I should sit down. An overweight coach and gym teacher with a bulbous nose, he asserted that ERAs could never be higher than 27, the number of outs in a complete game. For good measure he cackled derisively.

Later that season the Milwaukee Journal published the averages of all the Braves players, and since this pitcher hadn't pitched again, his ERA was 135, as I had calculated. I remember thinking then of mathematics as a kind of omnipotent protector. I was small and quiet and he was large and loud, but I was right and I could show him. This thought and the sense of power it instilled in me was thrilling. So, still smarting from my earlier humiliation, I brought in the newspaper and showed it to him. He gave me a dirty look and again told me to sit down. His idea of good education apparently was to make sure everyone remained seated. I did sit down, but this time with a slight smile on my face. We both knew I was right and he was wrong. Perhaps not surprisingly, the story still evokes the same emotions in me that it did decades ago.

So, is what this teacher did good pedagogy? Of course not, and happily I benefited from many more knowledgeable, supportive, and nondirective teachers and a variety of pedagogical approaches. Nonetheless, this particular teacher did give me a potent reason to study mathematics that I think is underrated. Show kids that with it and logic, a few facts, and a bit of psychology you can vanquish blowhards no matter your age or size. Not only that, but you can sometimes expose nonsensical claims as well. For many students this may be a much better selling point than being able to solve mixture problems or using trigonometry to estimate the height of a flagpole from across a river.

Not unrelated is another bit of mathematical pedagogy that was of benefit to me early on: board games, Monopoly in particular. In this game players roll dice to move around a board lined with properties (as well as railroads and utilities), which they can purchase with the game's money, develop by purchasing houses and hotels to occupy them, and from which they can derive rents from the other players who happen to land on these board spaces. The point of the game is the lovely one of driving one's opponents into bankruptcy. Like the best teaching, it's invisible.

For example, to determine how likely I was to land on Boardwalk or Jail I needed to figure out the probability of the various outcomes when rolling a pair of dice. Obtaining a sum of 7, I realized, was the most likely outcome, arising in 6 out of the 36 possible outcomes—(6,1), (5,2), (4,3), (3,4), (2,5), and (1,6)—whereas 2 and 12 were the least likely, each arising in only 1 of the 36 possible outcomes—(1,1) or (6,6). As anyone who has ever played the game knows empirically, Jail is the square on which players spend the most time. Thus the orange properties are good ones because they're relatively cheap and visited often by those leaving Jail.

My childhood Monopoly discoveries and competitiveness are not the point here. Rather, it is that implicit in the game, whether young players realize this or not, are a number of important mathematical ideas that one gradually absorbs while playing the game, in my case for untold hours on boring summer afternoons. Among these ideas are probability, expected value (average payouts per owned property), even Markov chains, which explain which squares are most likely to be landed upon. (The latter are systems like Monopoly that transition from one state—a player's place on the board, say—to another, the next state depending only on the present one.)

The lessons imparted by the journey around the Monopoly board are in some obvious ways relevant to the journey through life. Being cognizant of probabilities; being sensitive to possible playoffs, risks, and rewards; and being aware of long-term trends are all useful life skills.

The game, not to mention the vagaries of life, also allows for ad hoc and difficult-to-quantify changes to the rules. One such rule I remember is that if an adult enters the room when a player is rolling the dice, that player must pay a fine of $2,000. We soon abandoned this rule, however. It led to arguments when a player who was behind began suspiciously and loudly coughing when it was the leading player's turn to roll the dice. Related problems arose with the rule that allowed an occasional looting of the Monopoly bank.

Unfortunately board games are a bit passé today, but many (not all) of the same lessons and other quite new ones can be had via video games. In fact, mathematics professor Keith Devlin has advocated the use of appropriately chosen games as vehicles for imparting mathematical ideas to middle school players. Note that I used word players rather than students intentionally. The games generally are chases or fights in which the players must solve puzzles and devise strategies to best their opponents.

OF MOTHERS AND COLLECTING BASEBALL CARDS

An almost canonical story that straddles memoir and (in my case) math is about mothers who think it is their duty to throw away arguably childish but still valued possessions when one reaches a certain age. My experience confirms the cliché and illustrates a nice bit of mathematics. As a young boy I was an avid collector of baseball cards, and for a couple of years in the late ’50s I managed to collect the complete set of cards that usually came in packs of five along with a piece of pink bubble gum that I loved but would now characterize as revoltingly sugary. I remember how long it took to obtain the last two or three cards of the collection that I hadn't yet obtained. If memory serves, the last holdout was that of Charlie Grimm, who at the time was manager of either the Chicago Cubs or Milwaukee Braves, the acquisition of whose card required the purchase of hundreds of cards, all except one of which were doubles, triples, quadruples of cards I already had. After completing the set I put them in a little box, labeled them, and some years later went off to college. I've already revealed the punch line. Searching my drawers a while later, I discovered the cards missing and learned that my mother had thrown them out thinking I'd outgrown them. Since I rarely even looked at them, this was a reasonable but unsatisfying assumption. At the very least, had I held onto them into the age of eBay, I might have sold them for a nontrivial amount of money.

The silver lining to this prosaic story is that it got me thinking about a notion I vaguely understood but would soon learn mathematicians called the expected, or average, value of a quantity. In particular I wondered how many baseball cards on average one would need to buy before one obtained the complete set of, say, 400 different cards (making the dubious assumption that the companies printed the same number of each card). Probability theory2 says that to obtain all x cards, one would need to buy approximately x × ln(x) or x times the natural logarithm of x cards. (The natural logarithm is like the notion studied in high school algebra but with the base e, which will make several later appearances herein, rather than with the base 10.) This very nice result, which involves adding up a bunch of so-called geometric random variables, says that one would need to purchase approximately 2,400 cards, on average, to obtain the complete set of 400. No wonder I had more than my share of cavities in those days.

The same analysis applies to any set of items you obtain randomly and that have roughly equal probabilities of occurring. Stretching the conditions a little, you might try to say how long it would take you to achieve this or that collection of everyday adult experiences—stubbing your toe, having someone skip in front of you in line, losing some little item. Note that the analysis above doesn't apply to significant experiences you aim for, such as a so-called bucket list of places you'd like to visit before you die, since these experiences do not occur randomly. Even this situation, however, is subject to a similar though more complicated analysis. We can consider different probabilities of attaining each element on your bucket list, for example.

(A little explanation of the collection problem that is eminently skippable: let's check that, on average, you would need to roll a die 14.7 times before “collecting” all six numbers on it [or all six action figures in some brand of children's cereal]. The first time you roll the die you will, of course, get one of the six numbers. After this happens, the probability of obtaining a different number on the next roll is 5/6, and so it will require a bit more than one roll to get the second number. It can be shown that on average it will require 1/(5/6) or 6/5 rolls to get the second number. After this happens, the probability of obtaining a third different number on the next roll is 4/6, and so it will also require more than one roll to get the third number. It can be shown that on average it will require 1/(4/6) or 6/4 rolls to get the third different number. We continue in this way. After obtaining 3 different numbers, the probability of obtaining a fourth different number is 3/6, and so on average it will require 1/(3/6) or 6/3 rolls to get the fourth different number. Likewise, to get the fifth and sixth different numbers requires 6/2 and 6/1 rolls, respectively. Summing 1, 6/5, 6/4, 6/3, 6/2, and 6/1 gives us 14.7 rolls, on average, to “collect” all six numbers.)

My baseball cards were thrown out, but the math stayed with me. It was much more valuable anyway.

A FURTHER NOTE ON MATH, HUMOR, AND MY EDUCATION

In high school I was what would today be called a nerd, but I was considered by some classmates a cool instance of one. I both smile and cringe to recall how cool (such an uncool word now) I thought I was pantomiming the Everly Brothers and Elvis Presley with my imaginary guitar and swirling pompadour. (Selection bias on my part induced me to internalize the statements of these friendly classmates.) I dated hardly at all, and pornography—Playboy magazine was the closest approximation to such—was nowhere near as available as today; there was precious little real or virtual sex at Milwaukee's Washington High School. I did read a lot, however, and in the summer between my sophomore and junior years of high school I spent every morning under the tree in my grandmother's front yard in Denver attempting to review and copiously supplement everything I'd studied up to that time, including old textbooks, novels, biographies, essays. Montaigne was a particular favorite, as was the math columnist Martin Gardner and even the tabloid Rocky Mountain News. I remember thinking that I had established a solid foundation for future academic/intellectual pursuits and thus that that summer was somewhat pivotal. In retrospect those days, the mornings in particular, seem idyllic, safe, and anxiety-free, but as with all such memories I wonder if they really were as remembered.

High school is often described as a particularly turbulent time in a person's life, but this wasn't the case with me. I recall having three desires at the time: to learn, to get away from home, and to have sex. As I mentioned, I did manage to learn a lot before I went away to college.

At one time or another in high school or as an undergraduate at the University of Wisconsin in Madison I majored or contemplated majoring in classics, English, philosophy, physics, and, of course, mathematics. I loved Latin and reading Caesar and Virgil and diagramming the Latin sentences in them, even though the ablative absolute didn't lend itself to modern conventions. My interest in classics, however, waned over time, as did others.

Despite the brief separations and flings with the above disciplines and other topics, I gradually became more deeply enthralled with the beauty, elegance, and power of mathematics. One example that struck me at this time: randomly pick a number between 0 and 1,000, say, 356.174, and then pick another, say, 401.231, and add it to the first to get 757.405, and keep doing this until the sum exceeds 1,000. On average, how many random numbers need you pick for their sum to exceed 1,000? The answer turns out to be e, which is approximately 2.71828, a non-repeating decimal that is not the root of any algebraic equation; is, as noted, the base of the natural logarithm; lies at the foundation of compound interest, mortgages, annuities, and modern finance in general; and is as pervasive a celebrity in the community of numbers as its better known cousin pi.

My fascination with e has not waned. Recently I described in an article on ABCNews.com how the number e lurks even in the night sky.³ To see this, imagine dividing some square portion of the night sky into a very large number, N, of smaller squares. Find the N brightest stars in this large portion of the sky and count how many of the N smaller squares contain none of these N brightest stars. Call this number U. (I'm assuming that the stars are distributed randomly, so by chance some of the smaller squares will contain one or more of the brightest stars, others none at all.) I won't do so here, but it's not difficult to show that N/U is very close to the very same number e and approaches it more and more closely as N gets large. (A different analysis shows that pi also resides quite naturally in the night sky.)

Easily stated open problems also fascinated me (and still do). An example that incidentally also intrigued my son when he was in college is the so-called Collatz (3×+1) conjecture: start with any whole number. If it's odd, multiply it by 3 and add 1, but if it's even, divide it by 2. Follow the same rule with the resulting number and continue doing so with the succeeding numbers. The conjecture is that the sequence of numbers generated in this way will always end 4, 2, 1, 4, 2, 1,…For example, assume you started with 23. The sequence would then be 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1…. The conjecture has been checked with extremely large numbers, but it has never been proved.4

Why these unexpected and amazing connections that transcend all divisions of nationality, culture, gender, class, and time? Clearly their truth is not affected in the least by the personal characteristics of the people who discover/invent them. An exemplar of this universality is Srinivasa Ramanujan, the untutored Indian genius who contacted the British mathematician G. H. Hardy about his astonishing mathematical insights and theorems and traveled to England to develop them but died there at an early age. In Robert Kanigel's biography of him, which I reviewed for the New York Times, there is a very moving vignette of the boy Ramanujan alone in the shadows of a Hindu temple, his elbow dirty from repeatedly erasing his chalk slate, as he cogitated and calculated away.⁵

The universal beauty of mathematics seemed (and still seems) ethereal to me, its elegance was mesmerizing, and its reach appeared limitless. I remember wondering about the certainty of these sometimes mysterious mathematical truths. Might it derive from or be reduced to the laws of logic, and might mathematical statements just be circuitous ways of invoking the law of the excluded middle, “A or not A”? Or might mathematical truth be simply a matter of convention more complicated, but not any more inexplicable, than the fact that 27 cubic feet equal 1 cubic yard or 16 ounces equal 1 pound? Might it simply be a discipline governed by rules, much as the movement of pieces on a chessboard is governed by the rules of chess?

Or might mathematical truth be a reflection of the activities of numbers and figures cavorting in some sort of Platonic heaven? Is this where unfathomably large numbers—like 52! (52 × 51 × 50 × 49…× 2 × 1), almost equal to 1 followed by 68 zeroes, the number of ways to order a deck of cards—live and play? (This latter number is so mind-bogglingly huge that the order of any well-shuffled deck might rightfully be called a miracle since the probability of its occurring is almost impossibly, minusculely minuscule.)

Why was mathematics so useful? These questions, the actual theorems I was studying, and the ever-growing list of practical applications of mathematics both vexed and captivated me.

Another contributor to my enthrallment, I must admit, was the much earthier realm of humor. I'd always appreciated literal interpretations of figurative phrases (such as “this is only a fraction of what you'll pay elsewhere,” where the price is 5/3 of that elsewhere), self-reference, unusual juxtapositions and permutations, logical paradoxes, and incongruities of one sort or another, all elements of humor and, what is less well known, mathematics. In fact, as I've written elsewhere, ingenuity and cleverness are hallmarks of both humor and mathematics as is a Spartan economy of expression. Long-windedness is as antithetical to pure mathematics as it usually is to good humor. At the risk of being long-winded myself, I'll note that the beauty of a mathematical proof often depends on its elegance and brevity, qualities I prized even as an arrogantly dismissive teenager.

My cast of mind predisposed me to the study of logic and mathematics, which in turn furthered my natural attraction to such humor. I saw Groucho Marx's Duck Soup several times and even now remember exchanges from the movie. Minister of war: That's the last straw! I resign. I wash my hands of the whole business. Firefly (Groucho): A good idea. You can wash your neck too. And, as mentioned, I still like puerile jokes: Teacher: Johnny, name two pronouns. Johnny: Who? Me? And I hate unmatched (parentheses. Much later after moving to Philadelphia I even tried stand-up comedy, albeit rather unsuccessfully, and learned that my taste in “logical humor” was not widely shared. A corollary is that three minutes is a really long time with material similar to jokes that generally only mathematicians or logicians seem to like: “Spell ‘Henry.’ That's easy, HENRY. No, it's HEN3RY. The 3 is silent” or “I'd give my right arm to be ambidextrous.”

Although I dated some in college, where humor of a warmer sort was more of an asset than math, I spent most of my little free time with some good friends, a couple of whom I retain to this day, and continued my studious ways. The story of my collegiate years is devoid of any overt drama, although becoming acquainted with the Banach–Tarski theorem (math joke: the Banach–Tarski–Banach–Tarski theorem; the explanation is that theoretically a sphere the size of a tennis ball can be decomposed and reconstructed into a sphere the size of a basketball or even one the size of the sun)6 and Kurt Gödel's incompleteness theorems on the inevitable limitations of formal mathematical systems were exciting enough for me. I never did any drugs except for one silly attempt to smoke banana peels, which seemed like an appealingly natural way to get high. All in all, I loved Madison, and the names Rathskeller, Lake Mendota, Van Vleck, Picnic Point, and Bascom Hill still evoke pleasant memories.

People sometimes assume a conflict between enjoyment and hard work. If it's not pushed too vigorously, I think such a trade-off sometimes does exist. On the other hand, hard work, especially on a project, subject, or endeavor that one loves, is itself enjoyable. In any case, moving to the University of Washington in Seattle for my master's degree resulted in more of the same, the scholarly idyll interrupted only by the specter of the draft and the Vietnam War. Rather than face the prospect of fighting in a war I very much opposed, I decided to temporarily abandon my PhD studies and enter the Peace Corps in Kenya, where I taught math at Kakamega Secondary School and became acquainted with a world quite different than that of a graduate school in mathematics. (With more punch than sense, the innumerate might say 360 degrees different). I later returned to Wisconsin for my PhD in mathematics.