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First Date With Math
By Steve Sullivan
We go in search of actuaries’ eureka moment, the moment
“[Math is] a secret language.
Only very few people
My Dad was always good with numbers . Ace student at Cornell, a 90- day wonder in the Pacific in World War II, chemical engineer at Allied Chemical in Buffalo, N.Y. All of that was a piece of cake compared with trying to teach his son math. The battles were epic, the stuff of family legend. Victories on either side were Pyrrhic, at best. My father died before he could appreciate the irony of my current place of employment. But if you ever happen to hear somebody in heaven laughing his head off, that’s probably my dad. I wonder now, as I’m sure he must have wondered then: What genetic joke was being played on us that caused his mathematical gift to skip a generation? Why was something that came so easily and naturally to him so impossibly difficult for his son?
Rhonda L. Lessard, Hartford, Conn.
I think my first date with math, or at least the first I can remember clearly, was number squares. Remember those? An NxN matrix of numbers where each horizontal, vertical, and diagonal had to add up to some total, and you could only use each number in the matrix once. Is there anything more fun?
After that, math and I saw each other regularly. When I was young, my father would make up math problems for me to do in the evenings. He taught me negative numbers when I was 8 and on a third grade exam I put the answer to 3 – 5 as negative 2. My teacher marked it wrong; the correct answer was “Cannot Do.” My father was pretty angry.
As I got older, I would decompose numbers I came across into prime components, the way my mother would still see the Tetris blocks falling even after the computer was turned off. Of course, I never told people that I was thinking that way. It seemed weird then. Actually, it seems weird now!
Math and I were not exclusive; we both saw others in our “don’t ask don’t tell” relationship. In high school, I started secretly seeing his cousin, computer programming, and found him just as fascinating and sometimes even better company. He was much more complicated, and one conversation could keep me entertained for hours on end.
So in high school I had to start thinking about how this was going to work with me and math, whether we were in it for the long term, or whether I should just run off with his cousin. My father gave me an article to read about being an actuary—it had been rated the best career to have just the year before. Somewhere in the article was the matter of salary, and that pretty much sealed it for my 15-year-old self.
To my surprise, being an actuary has meant a long-term relationship with math’s whole family. And while at times they are a frustrating bunch, I can’t imagine life (or my wallet) without them.
Do Mathematicians Dream of English Class?
Those of us who are not gifted in math — who may indeed be mathophobic—tend to regard it as one of life’s great mysteries, a secret language on the order of Esperanto, say, or Klingon. We sat dumbfounded in classrooms as the math people spoke in tongues and scribbled hieroglyphs on the blackboard. These moments come back to us now as the stuff of nightmares—the final exam in the class we haven’t been to since the first day of school. Always a math class.
When we’re not reduced to panic, we tend to regard math with a certain amount of awe, even superstition. The way the uncoordinated regard the athlete who can actually hit a fastball. Surely there must have been a moment when the anointed realized they not only understood the language of math but were comfortable speaking it. What was that moment like?
I decided to investigate that question by posing it to a random sample of Contingencies readers and Academy members across the country. And by doing a little reading on the subject. What I came up with was surprising (at least to me).
David Bahn, Jacksonville, Fla.
Like most small boys growing up in the late 1940s, I became a baseball fan and loved to play the game. Southeastern New Hampshire, where I grew up, was, and still is, an integral part of Red Sox Nation. (Needless to say, I’m still savoring the 2004 series, the ALCS as well as the World Series.) The Braves were still in Boston, but they tended to be second to the Red Sox in popularity. By the time I was age 10, however, it became quite apparent that I was a fat kid who was not especially athletic. While my friends played real Little League, I barely made the “minors.”
On the other hand, I realized that my role in baseball was as a statistician, where I could really use my skills with multiplication and long division, whether or not anyone wanted the statistics calculated. In junior high, I even learned how to score the game. Writing “K” was much better than walking back to bench after making a “K.”
I continued with good grades in all subjects in high school, where I earned letters in baseball as a student manager, and in college, where I majored in math. So, here I am, an actuary who still loves the game, especially the Red Sox.
I was surprised that I was the only one in the class who got it and somehow understood how you could always guess the number without there being any magic involved.
Alice Rosenblatt, Thousand Oaks, Calif.
I first realized I was gifted in math in fifth or sixth grade. The teacher was reviewing a method for doing a puzzle. You would ask a friend to pick a number and then make up a series of adds and subtracts to that number. You would then subtract the starting number, and you could always tell the friend the resulting number.
I thought this was great fun and asked the teacher how it worked. The teacher explained that the number you started with should be thought of as “x.” Little did I know that I’d just had my first algebra lesson. I was surprised that I was the only one in the class who got it and somehow understood how you could always guess the number without there being any magic involved.
Steve Kantor, Atlanta
I had a very old and old-fashioned fifth-grade teacher. We were doing some subtraction problems, and she stated you couldn’t subtract a bigger number from a smaller number. I had a feeling that wasn’t quite true, and I soon realized she was trying to withhold the secret of the negative numbers from me.
Another math memory comes from high school. I was a good student by then, but my teacher gave me such a glow when I discovered an original proof for showing that the square root of two was irrational. As he said, it’s been done before, but it was still original to me.
Genetics or Gossip?
Not exactly the Eure ka moments I was hoping for. Instead, it’s more: “I always spoke numbers, the same way I always spoke English. What’s the big deal?” Almost as if it were ... genetic.
That’s why I was intrigued by a book that came out about five years ago called The Math Gene by Keith Devlin, a math professor and executive director of the Center for the Study of Language and Information at Stanford University. Turns out, however, that the title is a little misleading; he doesn’t really believe people are born with or without an actual gene for math. But he does believe we’re all born with the capacity to understand numbers, in the same way we’re born with the capacity to understand language.
But if that’s the case, why those nightmares? And why did I have to drive my poor father so crazy?
Here, in as much of a nutshell as I can get it, is Keith Devlin’s answer.
Math is the science of order, patterns, structure, logical relationships. What sets humans apart from other animals is the ability to recognize patterns, separate things into types, learn, and adapt. Homo sapiens became able to do this somewhere between 75,000 and 200,000 years ago when we learned how to use language. That enabled us to reason in a what-if, abstract way and to take our thinking “off-line.”
For Devlin, the “math gene” and the “language gene” are one and the same. The same mechanisms that enable most of us to understand language, almost by instinct, enable some of us to understand the abstract patterns of math. So that may explain why it’s so difficult for mathematicians to describe their eureka moment. It’s the same as saying to me, “Describe the first time you remember using language.” My parents might remember it, but I certainly don’t.
OK, that may explain why everybody has the innate ability to do math. But it doesn’t explain why not everybody can. The answer to that question, according to Devlin, begins with gossip.
Yes, you read that right. Gossip.
“Because evolution is not about making certain behaviors possible, it’s about making them easy and automatic,” Devlin writes. “What a few individuals may be trained to do doesn’t matter. Only what an entire species does easily, naturally, and by inclination, is evolutionarily significant. That a single elephant can be trained to push a baby carriage has no meaning for evolution. That over half of all human use of language is to talk about other people is very significant indeed.”
For mathematicians, then, math is gossip, as compelling to them as soap opera is to many millions of others. But instead of people as characters, the characters in the math soap opera are numbers, geometric figures, groups, topological spaces, even π. Mathematicians can see and understand the relationships among these mathematical abstractions the same way the rest of us understand As the World Turns. (Or not, as the case may be.) But according to Devlin, we all have that basic ability hard-wired into our brains.
What separates trained mathematicians (and by extension, actuaries) from the rest of us is threefold, says Devlin: the sustained concentration that makes it possible to be able to do math; the ability to visualize and work with abstractions; and perhaps most important, the simple desire to do it. “The key to being able to do mathematics,” he says, “is wanting to.”
Well, maybe so. I know I wanted desperately to get my dad off my case, but maybe that wasn’t motivation enough. And even Devlin admits he doesn’t have a definitive explanation for what makes some people comfortable in the world of mathematics and others run from it screaming. He maintains only that the door to that world is inherently open to everybody.
Or maybe not everybody. Linda Mallon, the admittedly math-averse editor of the Academy’s Update and EAR, speculates about the possibility of a math dyslexia.(Dyscomputexia? Dysnumeria?)
“We used to assume that people who had trouble reading were just slow or lazy,” she says, “until we discovered that it was a real learning disorder—dyslexia—that made reading supremely difficult for them. Maybe there’s a related disorder that makes numbers equally meaningless for some people.”
Keith Devlin, Executive Director, CS LI, Stanford University, Calif.
No eureka moment. Don’t see how there could be, really. Certainly my motivation for doing math was to become a physicist, and for many years I toiled away at math purely to use it, and it didn’t make a whole lot of sense as a body of knowledge in its own right. The whole show came together for me when I met calculus in high school. That’s the first essentially non-trivial math in the school system. Up to then, it’s really just formalized common sense. But calculus is cool and hard and, to all appearances, pure magic. It allows you to do things you would have thought impossible. That’s not when I realized I could do math. It is when I realized I would do math—as my primary interest.
Ruth Ann Woodley, Simsbury, Conn.
Before I entered the first grade, my father made the offhand comment that he expected me to be very good at math because my grandfather was. I took the prediction for granted and soon was the top math student in first grade.
In the second grade, I almost abandoned math when I struggled to memorize multiplication tables but then was hooked for good by the thrilling concept of “carrying” in addition.
I finished our assigned math workbook well before the end of that school year, and several later math teachers learned to let me spend the class time doing that night’s homework since I didn’t need the lectures. It was clear that Dad’s prediction was correct and math just came naturally to me.
Years later, I realized that many girls my age were subtly given the message that they should stay away from math and became afraid of it before realizing that they did have that talent. But I assumed that anything my dad said must be true. While I know now that I would have been good at math without that prediction, I give Dad credit for instilling a lifetime of confidence in me from that day.
And he has always been proud of my career choice; I often hear him bragging to friends that I’m in a field most people don’t understand.
The tempest has died do wn no w, but the atmospheric conditions linger. Back in January 2005, Harvard University President Larry Summers opened his mouth to give a speech on gender disparities and put his foot in it. In trying to explain the perceived gender gap among top-level science professors, he said that women aren’t as interested as men in sacrificing their lives for a high-powered job; men have more intrinsic aptitude for science; and women may be the victims of gender discrimination.
Mr. Summers derived his explanation after reading a book, The Blank Slate: The Modern Denial of Human Nature by Steven Pinker, one of his psychology professors at Harvard. But he must have skipped over a couple parts, including this one:
“The suggestion that the gender gap may arise, even in part, from differences between the sexes can be fightin’ words. Anyone bringing it up is certain to be accused of ‘wanting to keep women in their place’ or ‘justifying the status quo.’”
At night, Ryan puts himself to sleep not by counting sheep, but by doing times tables and exponentials. One day he’ll be a brillant mathematician, but for now he’s my brilliant 1st grader.
Nor did he note when Pinker quotes a female colleague: “Look, I know that males and females are not identical. I see it in my kids, I see it in myself, I know about the research. I can’t explain it, but when I read claims about sex differences, steam comes out of my ears.”
Summers released the steam by stepping squarely into the middle of an ongoing debate, the debate between what Pinker characterizes as gender feminism and equity feminism. Equity feminists acknowledge biological and neurological sex differences but are against sex discrimination in all its permutations. Gender feminists believe that men and women are essentially the same; the only differences stem from male oppression of and discrimination against women. What the gender feminists heard was the sound of a Harvard president once again slamming the doors of science and mathematics in their faces.
But is that what he was doing? Or was he, as Pinker maintains, merely expressing alternative views on why women might be underrepresented in the sciences?
“They have converted his suggestion that the statistical distributions of men’s and women’s abilities are not identical to the claim that all men are talented and all women are not,” Pinker wrote in the Feb. 14 New Republic, “as if someone heard that women typically live longer than men and concluded that every woman lives longer than every man.”
Though my sample of readers can hardly claim to be scientific, of the 19 people who responded, eight were female, 11 were male. That distribution isn’t reflected in the American Academy of Actuaries itself, which is about 20 percent women. Nor is it reflected in the 9 percent of women who are members of the National Academy of Sciences.
Of all the women’s responses, only one mentions her aptitude for math as an oddity, but not odd enough keep her from pursuing it. In the end, she paid no attention to whatever signals she was receiving about math being inappropriate for women. It seems to bear out Pinker’s assertion that “no sex difference yet discovered applies to every last man compared to every last woman, so generalizations about a sex will always be untrue of many individuals.”
To be clear, Pinker does not deny that women are often unfairly discriminated against and that this is one reason for the disparity. But it’s only one reason, and suggesting there might be others in no way discounts or disparages any of them.
If more women don’t pursue careers in science, engineering, and mathematics, he suggests, it could be because some of them don’t want to. And they don’t want to because, if the research is correct, men and women really are different. They’re not just blank slates waiting to be written on by society, or even male chauvinists.
Pinker presents a catalog of research, much of which has
been conducted by women, showing that the brains of men and
women work differently. Among the findings most relevant to
Again, this is not to say that all men and women are either-or; only that the data point to these conclusions, in the same way that data lead actuaries to certain conclusions about an uncertain future. It could be, of course, that proportionately more women than men responded to my inquiry because they’re more interested in sharing their thoughts on the subject—which only bears out the data. And if, as SOA surveys suggest, actuaries need to become better communicators, more women actuaries may be just what the profession needs to nudge it in that direction. Which, of course, doesn’t mean that male actuaries shouldn’t learn to be better communicators, too.
Stephen V. Gilmore, Charlotte, N.C.
My first knowledge of being good at math came when I was in sixth, seventh, and eighth grades. But it wasn’t merely numbers that caused me to like math. What I really liked was the structure, logic, and patterns that are behind it. Without these, math is merely numbers and “cookbook arithmetic.” It was for these more advanced aspects of math, not just the numbers, that I majored in math in college.
After college, I worked in probably the least mathematically oriented area of the actuarial profession: retirement plans. I did that for 25 years. Paradoxically, it was the non-mathematical aspects of my job I liked best: client contact and consulting, researching the legalities behind retirement plans, writing promotional and training materials, and training employees. And while I’m no longer in the actuarial profession (was downsized out), I have fond memories of my 25 years in it. My most noteworthy memory is being a speaker for two sessions at the 2000 Enrolled Actuaries Meeting.
Now I’ve come full circle, and I’m teaching math at the college level. But I won’t be nurturing any future actuaries, for the math I’m teaching is high school-level remedial math for those ruined by a system that reduces math’s structure and complexities to mere numbers and “cookbook arithmetic.” I’m doing what I can to undo the damage. It’s not an easy battle, but I will not give up the fight!
Tom Giambrone, Professor of Mathematics, Buffalo State College
Mathematics to me is the study of patterns. Algebra is patterns in number systems, geometry is the study of patterns in nature, probability and statistics is the study of patterns of chance, all in both real and hypothetical universes. But it wasn’t until I began to engage in the activities of mathematics that I understood its language. When you’re engaged in mathematics you naturally speak the language because algebra, for instance, isn’t just about factoring, it’s about describing patterns. Letting others see mathematics as patterns is what my career has been about.
Karen Love, Washington
While my decision to enter into a mathematics career is unexciting and as simple as “math is easy,” I wanted to tell you the story of my son.
Ryan just turned 6 years old. Last year, during kindergarten, he came home one day, and said, “Mom, ask me what’s the square root of 100.” So of course, I played along and asked the question. The answer, of course, was the perfect “10”. So I decided to go further. I repeated to ask him each square: 25, 1, 16, 49, 36, 4, 9, 64, 81, and he gave the correct answer each time.
Now, we all know they’re not teaching square roots in kindergarten nowadays. His older friends at after-care taught him about square roots, and he understood exactly what they were talking about. Ryan enjoys math. He enjoys being the first to answer the tough questions, and he enjoys being able to figure out the answers “without using fingers.” At night, Ryan puts himself to sleep, not by counting sheep but by doing times tables and exponentials.
I have no doubt that Ryan will one day be a brilliant mathematician, but for now, he’s my brilliant first grader.
Obviously , my own first date with math was a nonstarter. But back when I was a middle-school English teacher, I roomed with a guy named Tom Giambrone (quoted above) who taught math at the same school. He always maintained that I knew more math than I gave myself credit for. At the time, I thought he was crazy. Now I’m not so sure. He may have been way ahead of the curve.
Today, Tom’s job is to teach math teachers how to teach math. It’s probably too late for me, but if Tom can get his students to recognize in their own students the same quality he recognized in me—and to actually bring it out—the ranks of the actuarial profession should be well populated for a long time to come.
Steve Sullivan is editor of Contingencies and assistant director of publications at the American Academy of Actuaries in Washington.
Contingencies (ISSN 1048-9851) is published by the American Academy of Actuaries, 1100 17th St. NW, 7th floor, Washington, DC 20036. The basic annual subscription rate is included in Academy dues. The nonmember rate is $24. Periodicals postage paid at Washington, DC, and at additional mailing offices. BPA circulation audited.
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