The Lure of the Dark Side

Doug Szajda—University of Richmond

In the interest of full disclosure, I must confess that though I was once a mathematician, I have since gone over to the Dark Side—computer science. And like any self-respecting Dark Sider, part of my job is to entice others to follow my path. For an undergraduate math major, this translates sim­ply—if you truly want to experience the power of mathemat­ics, then, while there’s still time, take as many applied mathe­matics, computer science, and statistics courses as you can.

I know, your major doesn’t require you to take any statistics or computer science courses. Sadly, such programs still exist. And I understand that it’s comforting to live in the cocoon of pure math. Theory is clean. It is elegant. Yes, pure mathematics can be beautiful in the same way that great literature, art, and music are beautiful. Real-world math, on the other hand, is messy. Hypotheses are not always clear. Boundary conditions and transition phases complicate analysis. Models have to be carefully balanced between being simple enough to be tractable, yet sufficiently detailed that they accurately model phenomena. Dealing with this can be nasty business. But it’s what is required if you want to really use mathematics.

And there are at least two good reasons why you should explore real-world applications of math. First, you like math, and applied areas are where you’ll get to see some really amazing mathematics. In the corridors of my department (a combined math and computer science department), there are 45 AMS “Mathematical Moments” posters. These fliers, which in some math departments should be considered false advertising, depict problems or research areas where math plays a fundamental role. Topics covered include robotics, speech recognition, cell biology, protein folding, and even crime solving. Of this (admittedly unscientific) sample, only three posters discuss problems that might be worked on by a pure mathematician—and one of these is solving sudoku. On the other hand, the topics mentioned on the other 42 posters are most likely examined by experts in the techniques of applied mathematics, statistics, or computer science.

The mathematical techniques most often mentioned on these posters include statistics, dynamical systems, graph theory, mathematical models, pattern recognition, image analysis, differential and partial differential equations, linear algebra, combinatorics, and optimization. As a mathematics major, you’re not likely to see most of these techniques, even if you pursue a math Ph.D., although ironically, your non-math friends might very well be introduced to the basics of very useful topics in linear programming, graph theory, probability, combinatorics, and game theory in the non-major courses they take to fulfill their math requirements for graduation.

The second reason you should take more applied courses is that you likely have an interest in technology, and you live in a technological society. You use a computer and cell phone, probably own an iPod (if not several), and are surrounded by devices that are controlled by microprocessors. And let’s be honest: you probably couldn’t exist without them. Do you want to graduate without having even a basic understanding of how these work? Moreover, you live in a world in which you are bombarded by statistics. It thus behooves you, as a more technologically inclined citizen, to understand enough statistics to be able to see what statistical results really tell us—and also how they can be used in misleading ways.

In case you are inclined to dismiss the opinions of a Dark Sider, then perhaps you will listen to the Mathematical Association of America Committee on the Undergraduate Program in Mathematics, which recommends in its 2004 curriculum guide that mathematics programs should promote learning that helps students better understand the uses of mathematics. This is a refreshing change from the historical norm where applied mathematics was often viewed as a debasement of the Platonic ideals of pure math, and undergraduate programs were designed for the less than 10 percent of students who might have the desire and talent to continue their studies at the graduate level.

So, if you are fortunate enough to be a part of a program that has opportunities for engaging the applied side of math, you’d do well to take advantage. I can assure you, it’s more fun on the Dark Side.

The money isn’t bad either.

About the author: Doug Szajda is an associate professor of computer science at the University of Richmond. He is currently general chair of the Internet Society Network and Distributed System Security Symposium.

Aftermath essays are intended to be editorials and do not necessarily reflect the views of the MAA.

Does the Master’s Degree in Mathematics Get Too Little Respect?

Carl Cowen - Actuarial Science Program at Indiana University–Purdue University

If you think about the history of science, mathematics sits in a unique position: everything that has ever been true in mathematics is still true! We no longer believe that the elements are Earth, Air, Fire, and Water, for example, but Euclid’s description of geometry in the plane is still correct. Modern physics rests on developments from the late 19th century onward, with recognition that Newton’s discoveries provide a working foundation. Modern chemistry is largely a 20th-century science, and molecular biology starts with the discovery of the role of DNA in the mid-20th century. A fundamental difference between undergraduate education in mathematics and that of the other sciences is that we (mostly) take students to the early 20th century or so while the other sciences take students to the research forefront.

As an example, a few years ago I taught a course on computational neuroscience for juniors and seniors with a mathematical background or a biological background (prerequisites: two semesters of calculus for biology students; differential equations for math students; and at least junior standing in a mathematics, statistics, engineering, or biological sciences major. Note that no biology prerequisites were asked of the math students). During the semester, we read a research paper from 1988. The math students were astonished: they mostly had never seen a research paper, or if they had, they had never seen one that new! The biology students were also astonished: they had seen many research papers, but they had never seen one that old!

Thus, our science colleagues have a quite different perspective on undergraduate and graduate education than we do. A Ph.D. in chemistry at Purdue University requires two (two!) classroom courses, and the rest is research. A Ph.D. in mathematics usually includes 10 to 15 classroom courses! My own opinion is that the study for the Master of Science degree is the most intensive learning experience in the mathematical sciences. Much more mathematics is learned than at the undergraduate level because the study is so much deeper, and more is learned than at the Ph.D. level because there the learning is specialized and research focused. Thus, first and foremost, I regard the M.S. as the time when students acquire a broad and deep understanding of mathematics.

Further, most of the master’s program is devoted to studying late 19th-, 20th-, and 21st-century mathematics. Indeed, an M.S. program should put a student close (say 1950s–1970s era) to the research forefront in at least one area. Most M.S. programs include Ph.D. qualifier courses. This is fundamental, broad, and deep material in comparison to undergraduate work.

As a profession, we put too little emphasis on the M.S. and give it too little respect. We should be encouraging many more of our undergraduate students to go to graduate school and get an M.S. degree. Mathematics faculty are good at encouraging the “best” students to go to graduate school, but we should be encouraging the top third of our students to go on—they are surely qualified for the experience and would benefit greatly from the added education.

Moreover, the job surveys I’m familiar with suggest that the M.S. is the most marketable degree in the mathematical sciences. This is a consequence, I believe, of the fact that M.S. students know much more mathematics than undergraduates and are less likely than Ph.D.s to be “distracted” by research interests (in the minds of those who are looking for mathematical expertise in filling job openings).

There are several important career paths for M.S. degrees. The M.S. in statistics is the professional degree for a statistician. As I understand it, except for specialized areas such as the pharmaceutical industry where the Ph.D. is preferred, most “working” statisticians have an M.S. in applied statistics or biostatistics. The two-year college faculty member in mathematics is usually expected to have a “plain vanilla” M.S. in mathematics with enough statistics background to be able to teach beginning statistics courses. Both of these career paths are full of opportunities!

Aftermath essays are intended to be editorials and do not necessarily reflect the views of the MAA. To respond, go to Aftermath at www.maa.org/mathhorizons.

About the author: Carl Cowen is professor of mathematical sciences and director of the Actuarial Science Program at Indiana University–Purdue University at Indianapolis. He is a former president of the MAA. Email: ccowen@math.iupui.edu

Facebook and Texting vs. Textbooks and Faces

Susan D’Agostino - Southern New Hampshire University

Last semester, my business statistics students were not exactly thrilled when I announced an in-class ban on electronic devices, including laptops, phones, and digital music devices.

“But I use my cell phone as a calculator!” one student protested.

“Can’t I use my MP3 player to help focus during exams?” another pleaded.

“I found a cool app that gives p-values for the standard normal distribution!” another offered hopefully, as if using statistical jargon would entice me to cave.

“Humor me,” I responded. “Let this class be the one hour and fifteen minutes of your day in which you are completely unplugged.” I felt like a counselor at an outpatient program for recovering addicts.

Halfway through the semester, I did what any self-respecting statistics instructor would have done: I surveyed my 67 students and used the tools I was teaching—confidence intervals for means and proportions—to compile the data. My results provide estimates—with a 95 percent confidence level—for the in-class, electronic multitasking habits of business majors at midsized, regional universities. Every student in this category has, at some point, used a laptop, phone, or digital music device in class. In a seventy-five-minute class that permits students to be “plugged in,” a student with an open laptop takes electronic notes just as much as he social networks: 34 minutes with a margin of error of 5 minutes. Looking at websites that are relevant to class is only slightly more common than looking at websites that are irrelevant to class: 36 as opposed to 32 minutes. A student with an open laptop spends, on average, 27 minutes sending and receiving email and 11 minutes reading an electronic newspaper. That these numbers sum to more than the seventy-five class minutes hints at the prevalence of in-class, electronic multitasking.

Overall, when electronic devices are permitted in class, a majority of students using the devices—58 percent— multitask at least half the time. Students self-reported on the number of multitasking activities they engaged in beyond listening to the lecture or participating in class discussion: 52 percent of the examples involved one activity, including social networking or texting. Forty-six percent of the examples cited two, three, or four activities, including social networking, emailing, and doing homework. An intrepid 2 percent of the examples involved five multitasking activities: social networking, instant messaging, searching online, playing games, and texting.

To my surprise, the vast majority of students—94 percent—expressed either a favorable or neutral opinion of my policy. Were these the same students who originally made me feel like a counselor for substance abusers?

“Knowing I can’t text allows me to pay better attention,” wrote one student.

“Not having my computer out means that I can’t find myself on Facebook,” wrote another student.

“I like the reduced noise distractions from [the absence of] electronic devices,” wrote a third.

“It’s a good policy. I always see the students with laptops looking at Facebook or playing games,” another offered.

So what about the responses from students who did not appreciate my policy? One commented that he “miss[ed] the unlimited amount of information that a computer has.” Another was put off by having to “carry notebooks and pens for note taking.” Another mentioned his concern about being unreachable in an emergency. Of course, I had informed my students that the university’s security office would deliver an emergency message to a student in class if needed.

The Kaiser Family Foundation recently reported that the average 18-year-old spends over seven hours daily using electronic media devices for recreational purposes outside of the classroom. Based on my study, this statistic would likely increase dramatically if recreational use of electronics inside of the classroom were counted.

College students should not sell their in-class time short. Class should be a time and place devoted to wrestling with ambiguity, not deferring to online encyclopedias edited by anyone with an inclination to blog. Currently, this assistant professor of math is wrestling with whether the anonymous student who wrote the following comment on my survey intended to be ironic: “I think [the in-class ban on electronics] is a good policy.... In this age of technology, people need to stay connected at all times. It absolutely gets in the way during class. Unfortunately, I really do not know how to fix the issue. I guess you could Google it?”

About the author: Susan D’Agostino is an assistant professor of mathematics at Southern New Hampshire University.

Aftermath essays are intended to be editorials and do not necessarily reflect the views of the MAA. Contact information is available here.

The Intermediate Under-Valued Theorem

Bruce Peterson - Middlebury College

“Well, duh.”

A familiar student reaction to the Intermediate Value Theorem. After all, if a function is “continuous,” it can’t jump from place to place without stopping in between. Or,

“Real functions are like x²or sin(x). Those step thingies don’t really matter.”

This cherished theorem usually falls flat in beginning calculus because, I would argue, students see it as so obvious as not to merit discussion. And it’s not their fault; the theorem that justifies the word “continuous” strikes most students as unimportant because they rarely see it do anything other than confirm their long-held intuition about what “continuous” ought to mean. If a continuous function is positive somewhere and negative somewhere else then, sure, it has a root in between. But this familiar “application” is of course just a restatement—or a special case—of the original result.

So what kinds of applications are there? For starters, how do you cut a cake in half? What you don’t do is find the center and cut through it. Rather you mentally move a knife across the top until the area on the left of the knife looks to be about the same as the area on the right—a simple application of our old friend. Does the cake have to be round you ask? Nope. If S is any closed figure in the plane, then there is a line in any given direction that bisects the area of S. (A “closed figure” is a set bounded by a simple closed curve.)

To prove this claim, we can construct a standard coordinate system with the y-axis parallel to the chosen direction. For each x, let l(x) be the line through x and perpendicular to the x-axis. L(x), the area of S to the left of l(x), is a continuous function of x as is R(x), the area to the right of l(x). Hence D(x) = R(x) – L(x) is continuous. For a line to the left of S, D(x) = Area of S, and for a line to the right of S, D(x) = – (Area of S). By the Intermediate Value Theorem there is an intermediate line for which D(x) = 0 and L(x) = R(x).

If that were the whole story there would be no story. After all, we’ve really just beaten a simple theorem to a pulp and not learned much except that the Intermediate Value Theorem may be part of our DNA. Let’s look a bit further.

If S is a closed figure in the plane, then in fact there are two perpendicular lines that divide the figure into four “quadrants” of equal area. To see why this is so, let l(α) be a line that makes an angle α with the x-axis and, appealing to the previous argument, assume it bisects the area of S. Clearly l(α) and l(α + π/2) cut S into four quadrants. We’ll label them in the usual counterclockwise fashion and designate their areas A₁(α), A₂(α), A₃(α) and A₄(α). Since A₁(α) + A₂(α) = A₃(α) + A₄(α) and A₁(α) + A₄(α) = A₂(α) + A₃(α), we have at once that A₁(α) = A₃(α) and A₂(α) = A₄(α).

The difference D(α) = A₂(α) – A₁(α) is continuous, because each component is, and A₁(α + π/2) = A₂(α) and A₂(α + π/2)= A₃(α) = A₁(α). Therefore D(α) changes sign between α and α + π/2, and there is an angle for which A₂ = A₁ (=A₃ = A₄).

A better known example is the “Ham Sandwich” Theorem: Given a piece of ham and a piece of bread (in the plane), it is always possible to cut both in half with one slice of a knife. Intuitive? Obvious? The proof combines the ideas explored in the previous arguments—give it a try.

Here is a less familiar example: There is a square (not just a rectangle) that circumscribes any figure S in the plane in the sense that S lies inside the square and each side of the square contains a boundary point of S (possibly a vertex). To prove this one, let l(α) be a line tangent to S in direction α and with S on the left of l(α).The lines l(α), l(α + π/2), l(α + π) and l(α + 3π/2) define a rectangle R(α) circumscribing S. Let L(α) be the “length” of R(α), the dimension parallel to l(α), and W(α) the “width” of R(α), the dimension perpendicular to l(α). Since W(α) = L(α + π/2), applying the Intermediate Value Theorem to L(α) – W(α) proves the theorem. As you visualize the rectangle R(α) changing dimension, you can “see” the sought-after square.

The Intermediate Value Theorem won’t matter unless the instructor makes it matter, so here’s a final problem to ponder: Consider a planar set where the maximum distance between any two points is 1. Find the side length of the smallest regular hexagon that is guaranteed to contain any such set. (And be sure to check out the Zip-line section of The Playground in this issue.)

About the author: Bruce Peterson is Charles A Dana Professor of Mathematics and College Professor Emeritus at Middlebury College. His fondness for the Intermediate Value Theorem stems from a lifelong advocacy of geometry in general. He also has an avid interest in ornithology.

Aftermath essays are intended to be editorials and do not necessarily reflect the views of the MAA. Contact information is available here.

Thinking Inside the Box

Nathan Carter - Bentley University

I love computers and related gadgets, but have been wary about integrating technology into my classroom. Calculators are not allowed on most of my exams, my students and I use plenty of chalk, and PowerPoint rarely shows up. I graph by computer only if a hand sketch would be messy. But as the math world gazes with interest on a shiny, new WolframAlpha, formerly dormant debates over technology begin anew. And they got me thinking.

At first, I leaned right back on my old favorite argument, shared by many, that can be applied to many different pieces of high-tech math tools: “Technology is a black box that can actually get in the way of real learning when pushing buttons replaces a more rooted understanding of what’s going on below the surface. When used as a teaching tool, students may come away able to produce a few impressive answers, but they do so without real comprehension or the ability to apply their knowledge in any context other than the basic setting of the problems they’ve encountered in the assignment.”

There’s a lot of truth to this argument. Sure, some instructors might use it to justify a pre-existing preference—not wanting to rework the whole curriculum in response to a shiny, new WolframAlpha!—but that doesn’t mean the argument isn’t correct.

And I still think it is compelling, but I recently made an important realization. It’s also irrelevant. To see what I mean, let’s apply the same argument to a piece of mathematical technology that’s a little older than WolframAlpha, even older than the calculator—yes, even older than the slide rule. Let’s apply it to…algebra! (I’m talking quadratic formula and completing the square, not groups and rings.)

But is algebra a technology? Merriam-Webster defines technology as “the practical application of knowledge especially in a particular area.” The American Heritage Dictionary is less brief, but allows any “technical means” even if only from “pure science.” No sprockets or circuits are required! Algebra is a technology.

Why compare algebra, which takes so much thinking, to using a calculator or computer, which (often) takes comparatively less thinking? I suppose I could stave off this question by saying that in each case you must learn a technical skill, or you’ll make an error and thus get wrong answers. This is true, but there is a better answer.


Our old, faithful friend algebra has just as much potential to be a “black box” as calculators and computers do. This includes not only the too-common example of using algebra to derive just as ridiculously incorrect an answer as you might with a calculator, but it includes much more sophisticated missteps as well. Consider the mathematician who attacks a problem or a proof with all the metaphorical levers, buttons, and knobs in the algebra arsenal and comes out the other side victorious. Then the referee’s report points out a far-more-elegant, two-sentence argument. In such a case, the referee’s report might very well say, “The author clearly doesn’t understand what is going on in this argument.” Oh, what we miss by fleeing to the trusted algebra crutch too soon!

But isn't algebra useful precisely because it works even at times when we either don't know why or at least don't care to focus on why? Surely not every algebraic argument can be turned into elegant prose—at least not in short order. And more importantly, haven't we as instructors justified students' study of algebra for this exact reason—its utility?

If the power of algebra, when used rightly, to churn out correct answers from correct inputs is the reason that students should become proficient at it, then shouldn’t that same reasoning justify their becoming proficient at even more powerful tools? In fact, if we use that reasoning to justify requiring students to be proficient with algebra, how can we do anything but require them to be proficient with the likes of WolframAlpha? (Software engineers may now cackle and/or cheer.)

So this is how I saw the light. I am a new mathematics professor and I say that a mathematician who wants students to learn algebra should also want them to learn any similarly powerful mathematical invention, even if it has sprockets or circuits! Wait. This means that I have to rework my curriculum, doesn’t it?

About the author: Nathan Carter is Assistant Professor of Mathematics at Bentley University in Massachusettes and author of the acclaimed new book Visual Group Theory, which employs the graphic power of computers to explore abstract algebra.

Aftermath essays are intended to be editorials and do not necessarily reflect the views of the MAA. Contact information is available here.

A Politician’s Apology

Steve Kennedy - Carleton College

On September 10, 2009 British Prime Minister Gordon Brown issued a public apology, on behalf of the British nation, to Alan Turing for that country’s treatment of him. Alan Turing was one of the great mathematicians of the twentieth century. He worked on fundamental problems in mathematical logic in the 1930s in the process inventing some of the seminal concepts of computer science, including the basics of the theory of computation. During the Second World War he served at the British code-breaking school at Bletchley Park and was largely responsible for breaking the German U-boat code. After the war he worked on the design and construction of two first-generation computers and was an early advocate for, and theorist of, the future of artificial intelligence. Turing was also gay and in 1952 he was convicted of “gross indecency” and sentenced to a course of treatment with female hormones designed to eliminate his sex drive. Not surprisingly, this chemical castration had profound physical and psychological consequences. In 1954, at the age of 41, Turing took his own life.

The extraordinary drama and tragedy of Turing’s life—war hero and mathematical genius driven to suicide by persecution for being gay—has provided inspiration for several artistic interpretations of his life. (Turing did his best to increase the drama by choosing a method of suicide inspired by Disney’s Snow White, a favorite of his—he ate a cyanide-laced apple.) There have been at least three plays, one novel, and two movies based on his story, as well as an extraordinary biography by mathematician Andrew Hodges. That biography reveals Turing as a good and decent man more or less bewildered by the barbaric treatment he received from his countrymen. These retellings of the Turing story have contributed to Turing’s status as something of an icon in the gay community. His profoundly original scientific contributions—in computer science he is memorialized in Turing Machines, the Turing Test, and computer science’s Nobel Prize, the Turing Award—have similarly preserved his iconic status among mathematicians and computer scientists. In fact, the Brown apology was provoked by a petition drive organized by a British computer scientist, John Graham-Cumming, inspired by admiration of Turing and dismay at his treatment.

Once Brown issued the apology the news, and reactions to it, zipped around the Internet. Most folks believed that the apology was long overdue and constituted a genuinely positive development, especially as a small contribution that might help chip away at still existing homophobia. And, to Gordon Brown’s credit, he understood that this should not be just about Alan Turing. Turing is just one particularly egregious and notorious example; Brown explicitly expresses regret over the “many thousands of other gay men who were …treated terribly” and even recognizes the “millions more who lived in fear.” Most fans of Turing seemed pleased; one friend told me he felt “elated.”

I didn’t feel elated and I wondered if there was something wrong with me. Oh sure, I recognized that this was a good and necessary step, but I couldn’t help but feel that it was not proportionate. The British government, in the name of the British people, tortured this good and decent man (and thousands of others) because they disapproved of his sexual habits. Now, half a century later, they offer only words of regret. Maybe I’d feel better if Gordon Brown vowed not to rest until gay marriage was legal in Britain. Of course in Britain today the legal status of gays is a thousand times better than it is in the US, so maybe Brown could work on educating America? How about passing a heavy tax—the Turing Tariff—on all computing hardware and software imported into the UK from countries, like the US, that still discriminate against homosexuals by banning gay marriage? The proceeds of the tariff could be donated, in the name of Alan Mathison Turing, to the leading gay rights organizations in the exporting country.

I know that’s not going to happen. Nothing really dramatic is going to happen. Still, I admire John Graham-Cumming and the thousands who signed his petition and am grateful for their efforts. I’m also appreciative of Gordon Brown’s understanding and grace. I do feel badly that I’m unable to celebrate; a terrible maltreatment of one of my intellectual heroes is being publicly recognized as such. But it just seems that this goodness and benevolence—and maybe all goodness and benevolence—are slow and discreet and progress in tiny incremental steps while hatred and injustice bash and roar and wreak enormities. And we don’t seem to ever learn. Elated? No, I’m not elated; I’m just very sad.

Dispassionate Mathematics

Rick Cleary - Bentley University

I am coming clean. I do not have a passion for mathematics. And I don’t think that’s a bad thing. Here’s why.

I was recently reading an article about a friend who solved an interesting open problem and was rewarded with some well deserved publicity. This mathematician is quoted as saying, “I get a problem like this and I don’t sleep at night.” Articles about mathematics and interviews with mathematicians seem to always include comments like this. There is an indication of a passion for problem solving and a related inability to function in the rest of the world while the math question remains open. I do not recall ever reading an article where a mathematician discusses a recently solved problem and simply says, “Yes, that was a nice result. A good day at the office. That’s what I’m supposed to do. It’s my job.”


Does a person have to possess an extreme level of commitment to be a good mathematician? To choose mathematics as a major in college, should one feel that it is a calling? How about for success in graduate school in one of the mathematical sciences? Can a person forge a happy career in our subject without passion, or is something approaching devotion a vital ingredient?

I think in many circumstances we have, sometimes almost unwittingly, made passion a pre-requisite to entering the major, much to the detriment of the field. There are some of us who have had successful and rewarding careers in mathematics and related fields without having been on the high school math team, without taking the Putnam exam as undergraduates, and without losing sleep over problems. I enjoy mathematics, and I think a day at a math meeting with interesting talks is a great way to spend my time. But at the end of that day when I’m at dinner and a colleague grabs a pencil and a napkin and says, “Here’s a cute problem,” count me among those who try to change the subject.

Of course passion for mathematics can be a wonderful thing and it may be a necessity for the giant steps that move the field forward. The truly great mathematicians have that trait and I recognize its value. I enjoy reading mathematical history and I delight in accounts of the single minded tenacity shown in solving hard problems, and the euphoric feeling of triumph when successful. Anyone who has watched the wonderful Nova episode about Andrew Wiles and his solution of Fermat’s last theorem can see the great interaction between a scientific breakthrough and a personal victory. But tremendous devotion is demonstrably neither a necessary nor sufficient condition for building a successful career in the mathematical sciences. To use a baseball analogy, it might be necessary for a “Hall of Fame” career, but those of us making contributions as utility infielders have a place too.

Making passion a requisite part of our culture has costs. Here are a few of them:

Requiring passion discourages talented students from studying mathematics.
Recruiting students to mathematics is like a political party trying to decide whether to appeal to a committed base or expand a point of view to broaden participation. A common theme among colleges that have large and successful math majors is that they have put up the ‘big tent’ and allowed lots of students in. Does this hurt the quality of their top majors? It doesn’t appear to. For one thing it gives them enough students to offer required and elective courses more than once every other year!

Passion makes for poor advising.
Advisors who want passion as a pre-requisite often see graduate school as the only winning outcome for students with talent in mathematics. This runs counter to one basic goal of a liberal arts education, namely to encourage people to be thoughtful, adaptable and open-minded in career choices. Mathematics majors can go on to be wonderful accountants, machinists or landscapers and perhaps make unexpected contributions in those areas thanks to their point of view.

Faculty may quit research sooner than they should.
When faculty members insist that their research must be elegant, ground breaking and the result of a deep commitment, a common mid-career result is to give up completely. But there are myriad open problems in other disciplines where we can be of great help. There are plenty of opportunities for pedagogical research. There are consulting opportunities in business and industry. My PhD is in statistics, so finding these opportunities might be easier for me than for people in more abstract fields, but there are dozens of underpublicized success stories of people who make contributions in this more mundane way.

I feel fortunate to have an enjoyable, rewarding and I hope useful career in mathematics and statistics. My nonmathematical friends see me as a math nerd who sees the world differently than they do, and some of my mathematical colleagues probably see me as a slacker who doesn’t really ‘do’ math. That’s a balance I like.