Thursday, October 28, 2010

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

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:

Wednesday, September 1, 2010

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.

Thursday, April 8, 2010

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.

Tuesday, February 16, 2010

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.