Wednesday, April 2, 2014

Every Math Major Should Take a Public-Speaking Course

Rachel Levy—Harvey Mudd College

In mathematics courses we practice deep thinking, clear writing, and effective problem solving. Excellent public speaking complements these skills. As one of my students put it:

No matter what we all do after college . . . [we] will have to speak to people. Every one of us will have a limited amount of time that we can convince someone else to see our point of view.

A public-speaking course can help you develop a superpower: the ability to communicate to a live audience in a clear, compelling manner. Every mathematics major should take such a course. Comments in italics are from my students in Math Forum, our required public speaking course at Harvey Mudd College.

Unfortunately, standing up in front of a group makes us nervous. Our hearts beat faster; this throws off our body chemistry and can make us feel ill. We fidget, rock back and forth, make awkward hand gestures, or stand unnaturally still. Our body language, voice inflection, and gestures reveal our discomfort.
There’s something fundamentally nerve-racking about giving a presentation. The first day of Math Forum, we all attempted to describe that near-indescribable feeling of speaking in front of an audience . . . Here I was, in a class that I had dreaded taking since hearing about it my freshman year, thinking I was the only person that had these feelings, . . . and yet the dozen other people in the class shared this same feeling.

Practice is key to taming our nervousness and to making a successful presentation.
With each talk I delivered throughout the semester, my confidence only increased. For my first talk, I was a nervous speaker that feared the whole ordeal, unable to deliver my opinions with sincere confidence. In contrast, for my last 10-minute talk, I was completely comfortable and calm. I had even begun to enjoy interacting with the audience during the presentation.

We can learn a lot from watching other speakers—professors, renowned lecturers, and classmates. If possible, watch yourself giving a presentation.
In re-watching the video of my second talk, . . . I noticed I sometimes shifted my body weight from one side to the other. . . . In my [later] talk, I felt at ease, and this was evident in my posture.

Careful preparation is essential to a first-rate lecture. Speaking tasks often have a fixed, typically short, time allotment. In a public-speaking course, you learn to deliver a message within a given time and to pare your talk down to its essence, so that there is no wasted moment. Although there are many ways to construct a successful presentation, you’ll learn how to write a strong introduction and conclusion, and how to connect them with a logical flow of ideas.

Written mathematics can be expressed elegantly and efficiently with words and symbols, but in a presentation, written words and complicated mathematical notation are difficult to follow. A lecture benefits from an effective use of images, clever uses of color, and careful placement of choice information for each slide. Generally: Less is more.

These are a small sampling of the many skills you will learn in public-speaking class. If no such class is available, consider other options, such as a theater class or a Toastmasters club; research groups and math clubs can also provide opportunities for you to give a presentation. These experiences can help you deliver confident, compelling communication about any topic, including mathematics.
I got a glimpse at the sorts of strategies I’ll need to give good talks: approaching a topic from the eyes of someone who is unfamiliar with it, eschewing notation unless it is particularly elucidating, leaving out ideas that don’t support whatever central message I want to present, and many more. . . . I am happy I was able to share the ideas I find interesting with the rest of my Forum class. It was unbelievably satisfying to finally give a talk I could be proud of.


Rachel Levy is an associate professor of mathematics at Harvey Mudd College and editor-in-chief of SIAM Undergraduate Research Online (SIURO).

This article was published in the April 2014 issue of Math Horizons.

Sunday, February 23, 2014

Steven Strogatz on Mathematics Education


Patrick Honner: What are your thoughts about the state of math education right now?

Steven Strogatz: My thoughts are mostly based on my own instincts as a teacher and what I’ve seen of teachers I admire. I don’t know much about the constraints that practicing teachers face in high schools right now, so my opinions are fairly uninformed. But I do worry about math communication and teaching in general.

Can I give you my “I have a dream” speech?

PH: By all means!

SS: In my dream world, everyone would have the chance to be a teacher the way Mr. Joffray [Strogatz’s high school calculus teacher and the subject of his book The Calculus of Friendship] was a teacher. His job was to teach us calculus, but he had his own vision of how to teach it and he followed that vision. He was creative, and he put his personal stamp on the course for us. He trusted his judgment, and the school trusted him. He could teach us the way he wanted to teach us, and he was a great teacher.

This is a profession that should be revered. What’s more important than teaching? Why not let teachers teach creatively and inventively? So that’s my dream: a world in which teachers are given the freedom to teach the subject they’re supposed to teach, the way that makes sense to them.

PH: You have two daughters in school right now. Do you think they are being exposed to math in a positive way?

SS: No, I don’t. I worry that my kids are not falling in love with math because it’s being presented as lots of procedures that they need to learn.

It’s too fast. My eighth-grade daughter is taking algebra, and one day she’s doing word problems, like “find three consecutive odd numbers that add up to 123,” and the next day she was doing something I’d never heard of—literal equations.

It just struck me as unbelievable that we’re doing word problems in one night’s homework. Students should spend at least two to three weeks on word problems. They’re hard! Every old-fashioned word problem is being thrown at her in one night.

PH: And then it’s off to literal equations the next day.

SS: I can’t imagine what any kid is doing who doesn’t have a math professor as a parent. The whole thing looks crazy to me. I’m sure even my daughter’s teacher doesn’t want to do it this way. Something is really messed up.

PH: Should math be a mandatory subject for kids?

SS: I’m conflicted about it—I don’t know what to think. There are a lot of students out there who would love math but don’t know that. So they have to be exposed, or maybe even forced, to take math to realize they like it. But after a certain amount of that, it becomes clear to a student that they don’t want to take more math. We as a profession should think about this again.

PH: What math do you think all people should know?

SS: Some amount of number sense is essential—for example, to know what it means when the store says certain items are 20 percent off. If you don’t know what that means, to me, you’re not educated. I feel comfortable saying that every person should understand fractions. But after that, what? Does a person need to know what a polynomial is? That’s not clear to me.

What should a person learn, if anything, after arithmetic? That seems like a pretty interesting pedagogical question, and I don’t believe our current curriculum is the optimal answer. Algebra I and II are good subjects, but so is network theory. It would be nice if people could understand how Google works, for example; it’s not that hard.

There’s a lot of fun in math. Do we really have to teach such dead material? If we could get a cadre of
people who love math and who get it the way you get it or the way I get it—people who know what math is about—you don’t need to tell them how to teach. You just leave them alone, and it’ll be okay.



Patrick Honner is an award-winning math teacher at Brooklyn Technical High School. He writes about math and teaching at MrHonner.com and is active on Twitter as @MrHonner.



This article was published in the February 2014 issue of Math Horizons, along with more of Patrick Honner's interview with mathematician and author Steven Strogatz. Yet more of the interview is available online as a supplement.

Friday, November 1, 2013

Limits to Growth

Priscilla Bremser—Middlebury College

For your next mathematical modeling project, download “AP Program Size and Increments” from collegeboard.org. Using the number of Advanced Placement exams given annually from 1989 (463,644 exams) to 2013 (3,938,100), develop a model that describes the growth of the program. In your analysis, discuss possible adverse consequences of such growth. I can suggest one or two.

The College Board tells students that AP courses will help them “stand out in college admissions.” Guidance counselors, along with college admissions officers, advise students to take the most challenging courses at their schools. Dutifully heeding this advice, high school students rush through the mathematics sequence to get to calculus, often taking as many as six other AP courses before they graduate.

At the end of this frenzy, a number of bright, hardworking students have weak algebra skills, effectively neutralizing any advantage they might have earned. They may have placed out of Calculus I, but they are only marginally prepared for Calculus II.

Over time the AP program has shifted from being a way to meet the needs of a few students who are ready for a challenge to a de facto admissions requirement for many who may not be. Having used their AP credit to get into Middlebury, a number of our students try to take calculus again, saying “I know I got a 5 on the exam, but I didn’t really understand it.” If placement into advanced college classes is truly the main objective, then something is amiss.

Breadth over Depth


Mathematics majors have told me that they didn’t see an ε or δ until junior year of college. Their AP Calculus courses did not include the precise definition of a limit, upon which calculus stands. The College Board’s course description calls only for “an intuitive understanding of the limiting process,” followed by a list of topics so exhaustive that I’ve never seen a single college course cover them all. Apparently “rigor” and “challenge” lie in breadth, not depth.


Students in high school AP programs who love mathematics may end up with a weak conceptual understanding of their favorite subject. Meanwhile, students better suited to a different math course feel compelled to take AP Calculus to enhance their transcripts. Once they all get to college, their math professors have some students who earned 5s on the exam as well as others who scored 3 or lower (58 percent of those who took the AB exam in 2013). At a conference I heard one mathematician say to another, “We’re trying to figure out how to deal with students who have taken the AP.” Join the club.

Figure the Expenses


Why has this happened? At $89 per exam, some grumble that it’s all about money. Defenders of the program would point out that the College Board is “a not-for-profit membership organization.” Still, nonprofits exist to perpetuate themselves and seem to be taking a grow-or-die approach. The majority of students who took the Human Geography AP exam in 2013— 67,070 of them—were in ninth grade. Are that many 14-yearolds truly mature enough to take a college-level course?

There’s no going back to the time when the AP program was simply a way for well-prepared students to get advanced placement. Indeed, at my own institution, the faculty voted down a proposal to do away with giving course credit for high AP scores, choosing instead to limit each student to five such credits. Meanwhile, the College Board advertises the program as a way to “save on college expenses.” College may be too expensive, but this purported remedy blithely disregards the significant differences between high school and college.

For extra credit on the modeling assignment, use demographic data to estimate the carrying capacity of this system. What will growth rates look like in the coming years? At what costs?


Priscilla Bremser is a professor of mathematics at Middlebury College. Her interests include number theory, mathematics education at all levels, and appreciating the Vermont landscape on foot, bicycle, and skis.

This article was published in the November 2013 issue of Math Horizons.

Thursday, September 12, 2013

i Can't Stand It Anymore

Travis Kowalski—South Dakota School of Mines and Technology

Complex numbers are essential tools of mathematics, providing beautiful connections between arithmetic and geometry, algebra and trigonometry, number theory and analysis. Unfortunately, few people outside the cloister of trained mathematicians know this. I teach a course on complex analysis, and each time I am dismayed to find that, even after 15 weeks of demonstrating how the use of complex numbers fundamentally unifies most mathematical concepts learned in undergraduate studies, there is still a nontrivial subset of students who say, “That’s nice and all, Dr. K, but they aren’t real. They’re still imaginary numbers.”

By its very definition the lamentable word “imaginary” describes something that does not exist or is utterly useless. Of course, these derogatory implications were just what Descartes had in mind when he coined the term “imaginary number” in 1637. Two centuries later, Gauss advocated the term “complex number,” but Euler’s introduction of the symbol i means that, no matter whatever else we may choose to call it, the adjective “imaginary” will always be associated with the root of –1. Students know what the letter i stands for. To them, it is a number that is imaginary and therefore irrelevant.

It is a self–fulfilling—and sadly self–defeating—prophecy. And so, it is time to retire i.

If the previous plea of “pejorative prejudice” is a bit of a stretch (or at least, needlessly alliterative), allow me to strengthen it with a bone fide mathematical argument for retiring Euler’s chosen notation. In a standard presentation, the complex number \(a+b\sqrt{-1}\) is identified with the point (a,b) in the plane. The way complex multiplication is defined, the effect on the plane of multiplying by \(a+b\sqrt{-1}\) is exactly the same as left multiplying each point (written as a column vector) by the real matrix
a -b
b a
and so multiplying by the complex unit \(0+1\sqrt{-1}\) is the 90° rotation
0 -1
1 0

Whatever one wishes to label this matrix, the letter I is off limits because I always refers to the multiplicative “identity” matrix. This suggests that I should not be used for the complex unit. In keeping with a consistent lettering scheme, I should represent the complex number whose multiplication coincides with that of I, but that’s just the multiplicative identity—that is, I should denote the real number 1.

What would be a better symbol? Why not just dust off the old $\sqrt{-1}$ notation and use that? Unfortunately, this is a choice fraught with peril. Following the traditional algebraic “rules of radicals,” we end up with paradoxes like \[-1=\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)}=\sqrt{1}=1\] A better proposal comes from multivariable calculus. In the notation of ordered pairs, complex multiplication takes the form (a,b)(c,d) = (ac bd, ad+bc). Using this and vector algebra allows us to write any point in the plane as

(a,b) = (a,0)(1,0)+(b,0)(0,1)
=a(1,0)+b(0,1)
=ai+bj

where i=(1,0) and j=(0,1) are the standard basis vectors for the plane. Not only does this provide one more piece of evidence to support the claim that the symbol i ought to refer to 1, it also means (a,b)=a+bj. This looks exactly like the standard form of a complex number with the vector j standing in for \(\sqrt{-1}\). Even more compelling, note that

jjj = (0,1)(0,1) = (-1,0) = -1,

so j is indeed a square root of –1.

Consequently, we should denote the complex unit by j or, if we want to emphasize its role as a complex number rather than a plane vector, the italicized letter j. In fact, electrical engineers already use exactly this same letter j, although their prime motivation is that i is already reserved for current.

Our main motivation is that the letter j doesn’t stand for anything in particular, and it most certainly doesn’t stand for “imaginary.” The symbol simply denotes a complex unit, a number that multiplies against itself to yield –1. It is a blank canvas on which to paint the utility of the complex number system, effectively banishing the confusion and distrust of that other letter, which shall no longer be named.


Travis Kowalski teaches mathematics at the South Dakota School of Mines and Technology.

This article was published in the September 2013 issue of Math Horizons.

Monday, April 1, 2013

Mathematical Habits of Mind

Karen KingNational Science Foundation 


When I look back on my own mathematical education, I have many people to thank for helping me develop productive mathematical habits of mind. I remember walking to the car with my dad on a bitter cold day on the way home from kindergarten, and I just had to understand how you could do subtraction with regrouping. Instead of brushing off my pesky questioning (and I was pesky), he explained it to me, writing in the frost on the car window to illustrate the ideas. Some years later, Linda Agreen, my Advanced Placement calculus teacher, made sure that I understood why the fundamental theorem of calculus was fundamental, even though that was not going to be on the AP test. These habits of seeking real understanding were solidified in the mathematics department at Spelman College, under Etta Z. Falconer and her colleagues.

Building on the foundation laid by my father and my other mathematics teachers, I learned the mathematical habit of doggedly pursuing a complete understanding of ideas. I also learned how to recognize when my understanding was not complete and the reasoning skills to address the situation.

Unfortunately, too many students of mathematics, whether in college algebra or abstract algebra, do not possess these productive mathematical habits of mind. Instead, they have picked up some bad habits along the way: a tendency to look for the quick answer, a lack of persistence when the answer is not obvious, memorization over understanding.

Why do I keep referring to reasoning skills as “mathematical habits of mind”? Because I believe that if we start thinking about these unproductive practices as habits of mind, it opens up a different set of strategies for addressing the problem. When Al Cuoco, Paul Goldenberg, and June Mark introduced the concept of mathematical habits of mind (The Journal of Mathematical Behavior 15, no. 4 [1996]), it was a powerful concept for rethinking K-12 students’ learning of mathematics.

Habits are behaviors we engage in unconsciously, but they are the result of a long evolution of choices we make at a young age. Habits of mind evolve from the choices that we make about how to think about ideas. Thus, my dad’s early intervention was important. At 5 years old, I was still making choices about how to learn. So were my teachers—in elementary school, high school, and beyond.

But too few students develop the habits of mind needed for more advanced mathematical learning. Presented with a problem with no obvious example to follow, a poorly trained student might start writing things down or try some calculations with no real strategy in mind. Faced with the task of learning to write proofs, a person without sound mathematical habits usually attempts to memorize various arguments instead of re-creating them from their internal logic. These habits may have served them well previously, but no longer.

Habits reflect what a person is likely to do in a given situation, especially a stressful one such as taking a test, and habits are notoriously hard to break. Smokers know that continuing to smoke has a high likelihood of leading to cancer and other diseases, but that knowledge alone is rarely sufficient for those who are trying to quit.

With this in mind, we need to ask whether the way mathematics is currently taught reinforces bad habits of mind. Is it too easy to get by for too long using bad mathematical habits? And where did these bad habits come from in the first place? The likely answer is that there are some entrenched teaching habits in need of attention.

Thinking in terms of habitual behaviors conjures up powerful analogies. How might we change our approach to learning—and teaching—math if we labeled as “unproductive habits of mind” those methods that serve us poorly? Just like the person who finally replaces smoking with a healthier habit—or better yet, who never starts in the first place—we will all be better served with healthier mathematical habits of mind.



Karen King is the former director of research for the National Council of Teachers of Mathematics. She has been a member of the mathematics education faculty at New York University, Michigan State University, and San Diego State University. 

This article was published in the April 2013 issue of Math Horizons.

Friday, February 1, 2013

What to Expect When You’re Electing

Stephen AbbottMiddlebury College 



When the national election finally came to a merciful end in November, there was one universally recognized winner whose name did not appear on any ballot. In a stunning denouement, political blogger Nate Silver may have permanently altered the way elections are reported—and run for that matter—and he did so by staking his claim to the veracity of Bayesian statistics.


Like everything else in an election year, Silver’s story is nearly impossible to separate from its heated political overtones, but in this case it is worth a try. Not only was mathematics well served, but its objectivity emerged as a potential means for making headway into the political storms that lie ahead.

Nate Silver’s first statistical love was analyzing baseball, which he did successfully for a sports media company after college, but in the run-up to the 2008 presidential election Silver began applying his mathematical tools to political forecasting. In March of that year he started a blog called FiveThirtyEight and made a name for himself by correctly predicting the outcome of every state except for Indiana in the Obama-McCain race. With its star on the rise, FiveThirtyEight was picked up by The New York Times, just before the 2010 midterm elections. In anticipation of 2012, the Times signed Silver to a multiyear contract.


And this is where the plot thickens. In addition to being a first-rate statistician, Silver is also a self-professed progressive with ties to the Obama campaign. Thus, when Silver’s blog showed Obama with a comfortable polling edge going into the final weeks of the election, attacks from conservative pundits began to fly. Denigrating the messenger is standard procedure in elections, but Silver’s methods—i.e., his mathematics—also became fair game. An L.A. Times editorial characterized the FiveThirtyEight model as a “numbers racket.”


Referring to Silver, MSNBC’s Joe Scarborough proclaimed that “anybody that thinks that this race is anything but a toss-up right now is such an ideologue [that] they should be kept away from typewriters, computers, laptops, and microphones for the next ten days, because they’re jokes.”


Silver’s series of responses make for some pedagogically compelling reading. “There were twenty-two poles of swing states published Friday,” he wrote in a November 2, 2012, post. “Of these, Mr. Obama led in nineteen polls, and two showed a tie. Mitt Romney led in just one . . . a ‘toss-up’ race isn’t likely to produce [these results] any more than a fair coin is likely to come up heads nineteen times and tails just once in twenty tosses. Instead, Mr. Romney will have to hope that the coin isn’t fair.” Silver then goes on to give a razor-sharp explanation of the difference between statistical bias and sampling error and how one accounts for each in assessing uncertainty.


The FiveThirtyEight author’s mathematical rejoinders only agitated his antagonists, who vowed to make him a “one-term political blogger.” But on Election Day Silver’s model was correct for all 49 state results that were announced that evening. And what about Florida, which was too close to call for several days? Silver had rated it a virtual tie.


Predictably, this “victory for arithmetic” was quickly employed as weaponry in the red versus blue debate. This is as unfortunate as it is counterproductive, and here is why. If we can agree on anything in today’s political climate, it is the need for a more productive means of public discourse. If we ignore Silver’s political orientation for a moment, what we have is an illustration of how mathematics, in the proper hands, can provide an objective foothold when the partisan winds start to blow.


What could mathematics, and a mathematical approach that prioritized proof over punditry, contribute to our ongoing debates about climate change? The national debt? The relationship of gun laws to violent crime? What are the chances that some disciplined mathematical analysis might provide an objective first step in bridging at least some of our philosophical differences?


I’d rate it a toss-up.


Stephen Abbott is a professor of mathematics at Middlebury College and coeditor of Math HorizonsThis article was published in the February 2013 issue of Math Horizons.

Image by Randall Munroe (http://xkcd.com/1131/)

Thursday, November 1, 2012

Necessary Algebra

Paul ZornSaint Olaf College

I remember vividly the moment—and the room decor, the time of night, and the LP on the stereo—when my cousin Jon taught me algebra.

He and I, then seventh-graders, enjoyed those hoary old story problems (Al is twice as old as Betty; in seven years . . .) that once appeared in magazines such as Life and Look. I had concocted a simple strategy that one might charitably call iterative: Make any old integer guesses and tweak them as the errors suggest. What Jon first saw, and memorably pointed out, was that an unknown, say A, for Al’s age, can be manipulated as though it were a known quantity like one of my guesses.

What thrilled me then was the prospect of zipping through an entire genre of contrived puzzles. What amazes me still is the power of one simple idea: You can manipulate unknowns and knowns to solve equations.

That prescription seems a decent nine-word summary of what algebra does, even beyond the seventh grade. Jon and I got a preview, however dim, of an idea bigger and better than we could have suspected. Every student should encounter, and eventually own, an idea so simple and powerful. I’m convinced that almost every student has a fighting chance.

Is algebra necessary? 


So asked a provocative New York Times op-ed last July. In fact, the title is slightly misleading. Author Andrew Hacker, professor emeritus of political science at Queens College, doesn’t question algebra’s larger importance. He notes cheerfully that “mathematics, both pure and applied, is integral [Hacker’s good word] to our civilization, whether the realm is aesthetic or electronic.”

Hacker’s different but equally provocative question is how much “algebra,” that “onerous stumbling block for all kinds of students,” should be required in high school and college. His answer: Much less. And less of other mathematics, too.

Here “algebra” is in quotes because Hacker’s beef is not really with that subject in particular. Indeed, Hacker sees both “algebra” and existing curricula idiosyncratically. His examples of supposedly superfluous material—“vectorial angles” and “discontinuous functions”—are unlikely examples of “algebra” and even less representative of what is typically taught. And Hacker’s en passant endorsement of teaching long division (right up there with reading and writing) surprised me. He doesn’t acknowledge, or seem aware of, creative efforts to improve school teaching of “algebra” by teachers like those supported and mentored by, say, Math for America.

Hacker’s real curricular concern is broader than algebra: It’s the curricular complement of quantitative literacy (QL). He refers generally to “the toll mathematics takes” (my emphasis), not just to difficulties posed by algebra. In this sense Hacker’s three Rs proposal—require QR, but not “mathematics”—is more radical, and Philistine, than the article’s title suggests. But let’s concentrate on algebra.

Where he’s right, and wrong. 


Some of Hacker’s rhetorical targets are legitimate. Algebra can indeed be taught rigidly and applied ineffectively. (I remember the joy of solving algebra puzzles but also tedious hours of FOIL-ing quadratics.) Hustling high school students toward calculus sometimes pushes them too rapidly for effective mastery through prerequisite courses—including algebra. And Hacker, keen to avoid “dumbing down,” suggests some interesting applications of QL methods to such topics as the Affordable Care Act, cost/benefit analysis of environmental regulation, and climate change. (Whether such topics can really be approached without algebra is another question.)

As Hacker observes, few workers use algebra explicitly in daily life. (We all use it implicitly.) To infer that algebra can therefore vanish from required curricula is mistaken. Similar arguments might be made against history, the humanities, and the sciences generally, none of which is widely practiced in daily life. More important in curricular design than eventual daily use are broader intellectual values, which algebra clearly serves: learning to learn, detecting and exploiting structure, exposure to the best human ideas, and—the educational Holy Grail—transferability to novel contexts.

Transferability is undeniably difficult, as Hacker duly notes. The National Research Council agrees (see Education for Life and Work Report (pdf)) and indeed stresses the value of “deeper learning,” of which a key element is the detection of structure.

“Transfer is supported,” says the NRC, when learners master general principles that underlie techniques and operations.

Algebra is a poster child for deeper instruction. We should teach it. Students can learn it. 




Paul Zorn is a professor of mathematics at Saint Olaf College and currently serving as president of the Mathematical Association of America.

This article was published in the November 2012 issue of Math Horizons.