Math Horizon's Aftermath

Unduly Noted

2012-02-01T09:00:00.000-05:00

Tommy Ratliff—Wheaton College
When I opened the MathFest program in Lexington last summer, I took one look at the first page and nearly yelled out loud “NO! NO! NO!” The inside cover to the program contained an advertisement for an online homework system with the following example:

Find the derivative of y = 2 cos(3x − π) wit with respect to x.

The assertion is that a competing system marked this answer as wrong, but the advertised system identified the expression as correct, demonstrating its superiority. I assume that the intent is to show that the system can recognize equivalent, but not identical, algebraic expressions. What caused me to react so strongly, however, is that I would have also marked the given answer as wrong. The answer should have been

The parentheses matter! The expression sin(x) represents a function!

I should be clear: My irritation is not directed at this particular homework system as much as at the entire mathematics community for the sloppiness in notation that we tolerate, and even encourage, when dealing with trigonometric functions. You can pick up almost any calculus text, peek into almost any math classroom, or attend any number of talks at various MAA events to find a plethora of examples of trig functions lacking their parentheses.

Why do I think the parentheses matter so much? This is not just a pedantic preference on my part. The lack of parentheses represents an irregularity in notation that obscures the meaning of the mathematics. We often use a space to indicate multiplication, as in or 3 sin(x), so leaving off the parentheses hides the fact that we are using a trigonometric function. The confusion is compounded when we say that the derivative of “sine” is “cosine.” If we were to be consistent, this would lead to applying a distorted product rule to get

I have seen students struggle with this, even when they understand the intent of the original notation. They correctly apply the chain rule only to confuse the order of operations at the end because they did not put the constant multiple of 3 at the beginning of the expression:

After all, why should you apply the cosine function to the first 3 in the 3x but not to the trailing 3? If we always used parentheses to enclose a function’s argument, then there would be no confusion.

An even worse abuse of notation occurs in the location of the exponent when a trig function is raised to a power. I will never write sin²(x) for sin(x)² because the first notation leads to ambiguity when discussing the inverse trig functions. Since f^{− 1} (x) is the standard, consistent notation for the inverse function of f(x) , we also use sin^{− 1}(x) for arcsin(x). If we were consistent with notation, a perfectly reasonable calculation would be

Notice that I had to make a choice about the meaning of sin⁻² (x) in simplifying the expression—an impossible task! Should it be sin²(x)^-1= 1/sin²(x) or sin ^-1(x)² = arcsin (x)²? The point is that we should never have to make this choice! We should be taking the derivative of the inverse sine function. This is horrific. The bad notation allows at least three different interpretations of the expression

We in the mathematics community pride ourselves on the consistency and deterministic nature of our discipline. I think we do ourselves a genuine disservice when we use sloppy notation that requires another layer of interpretation to understand the intended meaning. The purpose of mathematical notation is to provide clarity and, ideally, to provide insight into the mathematics being notated.

Therefore, I implore you: The next time you use a trig function, please remember the parentheses, put the exponent on the outside, and never, ever write anything like sin3x²cos^-25x.

Tommy Ratliff is a professor of mathematics at Wheaton College in Massachusetts where he enjoys thinking about voting theory, building new science centers, and being precise in his notation.

Statistics à la Mode

2011-11-01T08:00:00.005-04:00

Meg Dillon—Southern Polytechnic State University

The last time I taught introductory probability and statistics, I turned in my grades and asked my department chair to take me off the course permanently. I’d spent some time working on a committee to update the course and we’d modernized it roughly to my taste, so my chair was puzzled. The best I could offer by way of explanation was, “I just hate it.” Then I went to France and taught their version of the same course.

My stint in France lasted three weeks. Essentially, I was substitute teaching and not looking for more than an excuse to be in the country for a while. My students were second-year engineering students, pretty much like my students at home. And like my students, the French students were a few notches below elite. While the similarities between my home university and my French university were comforting, the contrasts in the probability/statistics courses could not have been more jarring.

Anyone who has taught or learned in a U.S. mathematics department recently knows the typical introductory probability and statistics course. It involves an expensive, gassy textbook with lots of color pictures, word problems involving industrial applications, and charts to help students navigate problems. American students purchase the textbook and far too often, the ancillaries the bookstore peddles alongside the text.

At my home university, the chair has some difficulty finding mathematics faculty willing to teach the course. While I can’t speak for my colleagues, to me the course seems oddly estranged from mathematics. There is a section on probability, and we love that: the probability laws, the counting. It’s possible to trick out that section and get a chewier piece of mathematics into the act, but, by and large, the course is a hodgepodge of recipes, motivated by problems involving IQ testing, rhesus monkeys, salamanders, and the like. Regardless of the text, there is almost invariably a peculiar pair of caveats presented as from on high: Never accept the alternative hypothesis, and never say the probability is 0.95 that the mean lies in a 95% confidence interval for the mean. I dreaded teaching it in France.

The French course, though, was a different kettle of fish. No one expects French students to shell out money for books, so the course was based on notes produced by the instructor of record. The notes were spare and lacked attribution. They started with simple examples involving coins, dice, and lifetimes of electronic gadgets, what one would expect. The definition of sample space appeared on page one. (That was fast.) The definitions of sigma-algebra (Gasp! Are they joking?) and probability space (Is this a grad course?) appeared on page two. The course spooled out from there. Yes, it assumed more calculus than we do but mostly in the more interesting problems, and it treated testing and interval estimates in much the same way we do. No one was joking, and this was not a grad course: it was introductory prob/stats, in an unapologetically mathematical setting.

Statistics is possibly the most important course we teach in mathematics: for life and for cultural literacy, a basic understanding of it is essential. The high schools teach it, yet I’ve heard excellent high school math teachers express fear, if not loathing, of the subject.

An introductory probability and statistics course based on mathematics is missing, not just from the math education curricula, but from American soil altogether, as far as I can tell. While we teach these courses from bloated texts that avoid mathematics, we might seize the opportunity to teach a critical life skill—understanding statistics—through an exposition that glorifies its foundation in mathematics.

A big chunk of statistics courses in the United States are taught by non-mathematicians, outside math and statistics departments. By the looks of things, students can often get by on facility with software and a foggy understanding of principles. We still see many of these students in the introductory course, though. Could we do better there? Could we rope these students in with mathematical ideas, and could this happen anytime soon?

I don’t know, but I’m hoping to go back to teach in France next year.

About the author: Meg Dillon is a professor of mathematics at Southern Polytechnic State University in Marietta, Georgia.

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

If You Think You Know It, Try to Teach It

2011-09-01T00:00:00.002-04:00

Maggie Cummings—University of Utah

I am involved with a Math for America (MfA) project at the University of Utah that helps individuals with strong backgrounds in mathematics (typically a baccalaureate degree in mathematics) become secondary math teachers in high-needs schools. What has been extraordinary to me in this work is the gap between general mathematical knowledge and mathematical knowledge for teaching (MKT). This disparity has received significant attention in teacher education circles. (See, for example, Ball, Thames, and Phelps’s article, “Content Knowledge for Teaching: What Makes It Special?” in the Journal of Teacher Education 59[5], 2008.) The general theme of research in this area is that there is a difference between “doing” and “teaching” mathematics and that while teacher content knowledge is necessary for pedagogical knowledge and skill, the former does not guarantee the latter.

At the University of Utah, we are trying to develop a conceptual understanding of MKT at the secondary level and a means by which we might measure it. In particular, we are interested in identifying knowledge and skills that secondary mathematics teachers need but that are not necessarily possessed by those with degrees in mathematics. It may seem ridiculous to think that individuals with degrees in mathematics don’t know all the math they need for teaching secondary school students, but here are some concrete examples of where we see a gap:

How can you help a seventh-grade student mentally compute 35% of 80?

Why is the area of a trapezoid ½h(b1+b2)?

Is there a geometric reason that the slopes of perpendicular lines are negative reciprocals of each other?

How might you explain why 5 minus -7 equals 12, or why the product of two negative numbers is positive?

Why is anything to the 0 power equal to 1?

When our MfA fellows begin the program, most can provide the algorithms or rules related to the above topics, but when pressed, they generally are not able to give student-friendly explanations that connect a tangible model to the algorithmic procedure. For example, fellows usually set up a proportion to solve the problem of finding 35% of 80 (x/80 = 35/100), but do not know of any way to make this problem simple enough for students to compute the answer in their head. (One approach is to understand that 35% is three and a half groups of 10% portions. Ten percent of 80 is 8, so three of them would be 24 and another half [4] would give 28.)

Indeed, once they examine the conceptualization, the models are not just intuitive—they actually enhance prior understandings of our fellows. The issue is that (a) these conceptualizations are vital to the work of teaching mathematics and (b) they do not seem to be developed in conjunction with typical preparation in mathematics.

It is not enough for a secondary teacher to say a negative times a negative is a positive—she must also be able to engage students in understanding why this is the case and then how this logic can be applied to other situations. In a similar vein, it is not enough that a teacher knows that a student made a mistake in simplifying an algebraic fraction; he must also be able to identify what the student was thinking in the erroneous simplification process. That way, the teacher has a better chance of helping the student connect his or her understanding of numeric fractions to algebraic fractions.

As we prepare individuals with strong backgrounds in math to become teachers, what we have learned is that advanced content knowledge in mathematics must be deliberately linked to content-specific pedagogical knowledge and skills. If that linkage is not made, advanced content knowledge stays “siloed” in the instructor, where it doesn’t do the instructor or the students much good.

Individuals wishing to teach mathematics at the secondary level need more than a strong background in advanced mathematics; they need a strong foundation in the mathematics they are going to teach. So, while it is essential that secondary math teachers understand abstract algebra, it doesn’t necessarily translate into the ability to teach basic algebra. If the truest test of understanding an idea is being able to teach it to someone else, then even some of the strongest graduating mathematics majors still have much to learn about the foundations of their chosen subject. The more they are willing to learn, the more their future students will be likely to follow suit.

About the Author: Maggie Cummings is an instructor with the Center of Science and Mathematics Education at the University of Utah. Email: margaritacummings@gmail.com

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

The Problem with Problem Solving

2011-04-01T00:00:00.002-04:00

Andy Liu—University of Alberta

There are two stages in solving a problem. The first stage is to convince yourself that you have solved the problem. The second stage is to convince other people that you have solved the problem. The first stage is the creative one and is indicative of how mathematics is actually done. The second stage is more formal and often has little in common with the first stage, but ironically it is how mathematics is usually communicated and taught.

Let us illustrate with a simple geometry problem.

Problem.

P is any point inside an equilateral triangle ABC. Perpendiculars are dropped from P to BC at D, CA at E, and AB at F. Which has the greater total area: triangles PAF, PBD, and PCE, or triangles PAE, PBF, and PCD?

The symmetry of this problem compels us to jump in with both feet and say, “They are the same!” However, the proposer of the problem may be having fun with us, so let’s test our hypothesis with some special positions for P. Putting P at the center of ABC, then at the midpoint of BC, and then coincident with A, we see that in each case our hypothesis holds true. (See figure 1.) So we are confident that our conclusion is correct.

Figure 1

A good way to solve problems is to make use of special cases. The first two attempts at incorporating the special positions into the general diagram are not particularly fruitful. The shaded regions do not correspond exactly. (See figure 2.)

Figure 2

The third special position—the simplest one—delivers! The shaded regions correspond exactly. Moreover, instead of having P coincide with A, we can have it coincide with B and C as well, yielding the diagram in figure 3.

Figure 3

Since PUV, PWX, and PYZ are all equilateral triangles, PZ and PW are both parallel to BC, so that P lies on ZW. Similarly, P lies on UX and VY. Moreover, PXAY, PZBU, and PVCW are parallelograms, and a diagonal of a parallelogram bisects its area. Everything falls into place!

We are now at the end of the first stage—which is where the trouble begins. If you read the solution to this problem in some problem book, you are likely to get the following:

Construct lines ZW, UX, and VY through P, parallel to BC, CA, and AB respectively. Then PUV, PWX, and PYZ are equilateral triangles while PXAY, PZBU, and PVCW are parallelograms. Denoting the area of a polygon Q by [Q], we have

This is very neat and very impressive. The key step is in the first line—the construction of ZW, UX, and VY. But notice that in our work on the first stage, this construction emerges as practically the final step. The two stages run in essentially opposite directions!

Emphasizing the second stage over the first has significantly negative consequences. Performing an excess of exercises may lead students to look at a problem and say, “This one I can do,” and start their write-up before thinking through their own reasoning. Alternatively, they may say, “I have never seen anything like this before,” and move on for lack of any ideas.

The most difficult thing about solving a problem is getting started. Where do ideas come from? There are many techniques, but no sure-fire method because problem solving is not a science. It is an art. Progress comes only with regular practice and sustained effort on the first stage of the process. Unfortunately, only reading solutions by other people is ineffective because other people tend to write only about the second stage of problem solving.

Perhaps this is to be expected. In the world of research mathematics, only the second stage is required to get a paper published. The paradoxical result of this is that we disseminate new theorems of mathematics in a fashion that offers little insight into how they were created.

Bringing this back to the world of the classroom, the upshot is that more attention needs to be paid to the first stage of problem solving. Perfecting the expository skills is certainly important; however, there will not be anything to write about if students cannot conduct informal explorations. They must learn to make something out of nothing, which is the essence of research.

About the author: Andy Liu teaches at the University of Alberta, from which he obtained a Ph.D. in mathematics and a graduate diploma in elementary education, thus becoming perhaps the only person officially qualified to teach mathematics from kindergarten to graduate school. Email: aliumath@telus.net

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.

The Lure of the Dark Side

2011-02-11T14:05:00.003-05:00

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?

2010-10-28T11:11:00.001-04:00

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

2010-09-01T00:00:00.004-04:00

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

2010-04-08T14:47:00.006-04:00

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

2010-02-16T14:53:00.005-05:00

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

2009-11-11T17:04:00.024-05:00

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

2009-09-23T12:46:00.010-04:00

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.

Graphical Dysfunction

2009-04-01T13:20:00.005-04:00

Frank Swenton - Middlebury College

The function is well known as one of the most fundamental concepts in all of mathematics, and it is equally well known as one of the most misunderstood by students in mathematics courses. Some would say that this is simply the nature of the student, and others would blame the advent of the graphing calculator. But the root cause of this problem lies much deeper, its seeds planted more than 300 years ago: it is the graph itself, which through the primacy it has attained as a means of dealing with functions, has obscured—and even supplanted—in the minds of students the very concept it was meant to illuminate.

In the Beginning, there were the Domain and the Range, each a distinct set in its own right, and there was the Function sending each element of the domain to one element of the range. Domain was domain, range was range, and never the twain did meet. Fast-forward to the 17th century: René Descartes rotates the range by 90 degrees and clanks it down right on top of the domain; upon these axes he draws a function’s graph: a construction both revolutionary and visually seductive—in fact, enough so to eclipse the function concept itself.

Considering functions via their graphs is a practice so deeply ingrained in the mathematics classroom that it escapes the critical eye given to all other aspects of mathematical pedagogy. Graphs readily provide easy answers (in some cases), but the answers are all too ready, too easy, and—most importantly—quite often incomplete or deceptively facile. X has become synonymous with “input,” and y with “output”; the derivative means “slope”; the integral means “area”; continuity means that a graph can be drawn “without lifting the pencil.” Even the very heart of being a function has sadly devolved into the “vertical line test.”

The fundamental problem is that these rough and ready geometric answers often serve to distract the learner from well-warranted further consideration of the true nature of each concept; they provide just enough of a simple, soothing “answer” with which to be content and to quiet any call for further thought. Moreover, when a learner is presented time and time again with graphs, it becomes far too easy to see only the graphs. When applying a function f to a value, the eyes quickly move directly to the point of the graph at the correct horizontal location, and there they stop. Seldom does the eye ever stray to the domain, at which the action starts, or the range, at which the action ends—the function’s meaning as the link from one to the other is lost, all attention drawn by the immense visual pull of its graph.

The effects of this graph-centric perspective on the function are felt far beyond the topic of the function concept itself. Inverse functions have students immediately flipping the plane or wrenching their necks; that inverses are simply rules sending elements of the range back whence they came becomes secondary. Differential approximation becomes the circuitous unraveling of a triangle tangent to a graph, finally arriving at a formula that is a direct consequence of the definition of derivative. The epsilon-delta definitions of limits and continuity become a crisscrossed clutter of horizontal and vertical lines on a graph, yielding a similar tangle in the learner’s mind, when the domain and range have no business intersecting in the first place. Careful inspection across the gamut of function-related concepts shows that graphs very often complicate or confound in their effort to simplify and explain.

Is the graph logically inconsistent? No. But we must reply equally in the negative to the question of whether a graph properly expresses the function concept in its entirety. One might rebut that the modern definition of the function from X to Y is as a particular subset of the Cartesian product X x Y. However, consider the logical definition of the implication “if P, then Q”: do we properly conceptualize it as “Q or not P,” or is this merely the logical statement that defines its formal meaning? A function, while encoded as a subset of X x Y, is conceptually no more a subset of X x Y than a poem is ink on a page; the difference is that no one forgets that a poem must be read aloud (at least in the mind) to be fully appreciated or understood.

A function acts: it sends each element of its domain to some element of its range; it maps subsets of the domain to their images in the range; it even pulls back sets in the range to their preimages in the domain. A function acts—and a graph simply sits, dead. We must view that static graph as a machine ready to act, or else the graph only serves to gloss over the concept; we must make the crucial effort not to be content with the limited perspective on functions afforded us by graphs alone. The graph is only a valid tool for studying a function when viewed not as the function itself, but as a representation of the function—only in vigilance of this crucial distinction will the concept retain its integrity; and with a healthy independence of graphs, the function concept can properly grow and flourish within the minds of students.

On Measurement

2009-02-10T17:06:00.002-05:00

Tom Tucker - Colgate University

What is the population of the United States? How far is it from my house to my office? How much is the national debt? All of these questions have numerical answers, sort of, but we all know it would be absurd to answer 307,285,671 people or 1135 feet, 11 inches or $11,245,734,298,635. First, none of these questions is well-defined. For the population of the US, at what instant are we talking about? Are we including US citizens who are living abroad? For the distance from my house to my office, where in my house? Where in my office? As the crow flies (an expression that should tell you we are in trouble here)? For the national debt, again at what instant? Second, even if the quantities were well-defined, they are all measurements subject to error. For most measurements, you’re lucky if you can get an error as small as 1 part in 100.

A few years ago, I served on a committee to revise the K-12 Mathematical Standards for New York State. I had followed the NCTM Standards since the late 1980s and there was one “strand” that always bothered me: Measurement. In the early grades, there were recommended activities with rulers, such as measuring the height of a desk. By grades 6-8, measurement consisted mostly of formulas for areas and volumes. By grades 9-12, the strand petered out. It was always the thinnest of the five strands. As I sat through the committee meetings, it occurred to me that it should be the thickest. My rule is this: If it has units and an error, then it’s a measurement. By that rule, every number of importance in our lives is a measurement.

Mathematics education in the US has done pretty badly on the matter of units, which is mostly ceded to science education. What a shame, since units are enormously helpful in understanding equations, notation, and terminology. Although it now seems generally accepted that mathematics should be taught from the algebraic, graphical, and numerical viewpoint, there is a fourth medium for presenting mathematics: words. And units are words, which reach students in ways that algebra, graphs, and numbers cannot.

The real failure of mathematics education, however, has been its treatment of error, especially relative error. Is an error of a foot a big error? Yes, if you are measuring my height, no if you are measuring the elevation of Mount Everest. Mathematicians know that only relative error makes sense. But relative error is almost nowhere to be found in the mathematics curriculum.

I was on the Mathematics AP Committee for the College Board when calculators were first allowed on the exam. We had to decide how much accuracy we required. Of course, any scientist would give an answer in terms of significant digits. Because we knew, however, that the concept of significant digits was not a standard part of the K-12 mathematics curriculum, we said instead “three digits to the right of the decimal point” (no mention of floating point). A year later, we wrote an exam problem on US soda consumption that entailed numbers in the billions of gallons. If a student chose gallons as their units, rather than billions of gallons, the required answer had 14 digits, more than a calculator could handle at the time. I always thought a clever student should have chosen quadrillions of gallons as her unit and given the correct answer of 0.000. The AP Committee had been painted into a ridiculous corner by the failure of the US mathematics curriculum to deal with relative error.

Estimation skills may sometimes be viewed as just another form of “fuzzy math,” even though when we grow up, we find in our everyday lives that there are two kinds of arithmetic: what we do in our head and what we do with a calculator or computer (everybody who does their tax forms with pencil and paper only, please raise your hands). But I am talking about the role of estimation in measurement, not arithmetic. A proper approach to measurement would also involve estimation. Physicists have long played the Fermi game of trying to estimate some strange quantity—the real estate value of Hamilton NY, the number of people who have ever played Major League baseball, the number of piano tuners in Chicago. All involve seat-of-the-pants estimates based on a few facts, like the population of Chicago. But why aren’t math students playing this game? Why aren’t 6th graders being asked about how many ice cream cones their school eats in a year? Why aren’t 9th graders being asked by their math teacher to go home and come back tomorrow with their estimate of the number of gallons of gas consumed annually by cars waiting at red lights, with a full explanation of their answer?

I understand that the other strands—number and operations, algebra, geometry, data analysis and probability—are important, especially for the scientific infrastructure of the US—but I would feel a lot more comfortable about the public understanding of the costs and benefits of our country’s policies if I knew we could get the measurement strand right.

NCTM Mathematics Standards