Wednesday, August 23, 2017

Sick of Viral Math

By Ilona Vashchyshyn and Egan J. Chernoff

 Figure 1. The "fiendish test."
People love to argue, a fact that is perhaps nowhere better demonstrated than on the internet. Case in point: The Dress. Some saw the dress in the infamous photo, which took the internet by storm in 2015, as clearly blue and black, while others saw, obviously, white and gold—and both sides felt it was their duty to (LOUDLY) proclaim their conviction to the world.

If there’s another thing people love, it’s hating mathematics. Maybe you’ve experienced this phenomenon firsthand: “Ugh, I haaaaate math. I can’t do it to save my life!” an acquaintance at a party happily chirps, rolling his eyes up to the ceiling after you admit that you are a mathematics major.

And yet, somewhat paradoxically, many of these same people eagerly engage with viral math problems on social media. Otherwise reasonable adults seem to forget their hatred of mathematics and argue vehemently about the answer to an arithmetic problem (“9-3÷⅓+1 is nine, not nineteen—you moron!”).

These problems, and the reactions to them, reveal several prevalent misunderstandings about mathematics. They suggest that, contrary to our prior assertion, people don’t really hate math. Yes, many may think that they can’t do math to save their life—but perhaps this is only because their school experiences shed little light on what mathematics really is and what doing mathematics really means.

A Case of Viral Math

To illustrate our point, we detail five prominent misconceptions about mathematics through the so-called fiendish test, shared by the Daily Mail’s Shivali Best on January 27, 2017 (“Can YOU solve this McDonald’s maths puzzle? Brainteaser that has left the internet baffled is harder than it looks,” http://bit.ly/ViralMathProb).

See figure 1 for our reproduction. We encourage you to work out the answer before reading on. But first, a warning from the puzzle’s creator: This problem is “only for geniuses” . . .

Myth 1. Math is just a bag of tricks.
You say the answer is 25? We will let a commenter respond: “Wow, just . . . wow. Pictures are too difficult for you? Tell me something. In the One burger plus chips 9 equation, how many individual packets of chips do you see? Look at the picture carefully, I know this is hard for the simple minded, but do count the number of individual packets of chips.”

Indeed, there is only one packet of French fries in the last equation, not two—a realization that may cause you to revise your answer. But even if you obtained the correct result, you may doubt your work, as the following commenter did: “15, I suppose, but there is always some little trick.”

This problem is thus more than an exercise in algebra. It confirms your long-held suspicion that mathematics is nothing but a bag of tricks designed by deceiving magicians, for the sole purpose of making you feel stupid. Or, as another commenter put it, “basically . . . mathematicians are [expletive]s.”

Myth 2. Math is memorizing a set of rules.
Maybe you caught the French fry trick, and you worked out the answer to be 60. In this case, “Congratulation, you failed preteen maths. Learn your order of operations. Multiply BEFORE addition.”

If BEDMAS (or BODMAS, BIDMAS, PEMDAS, or PEDMAS, depending on where you’re from) wasn’t carved into your brain in the fourth grade, you may have indeed forgotten that in an expression such as multiplication takes precedence over addition.

Although you used careful mathematical reasoning to determine the values of the variables in the first three expressions, carefully juggling several values in your head until you deposited them into the last equation, it was all for nothing, because math is not about reasoning: It’s about BEDMAS; FOIL; Why ask why? Just flip and multiply; . . . “Didn’t you learn ANYTHING at school???”

Myth 3. Math problems have only one right answer.
Perhaps you are entertaining the notion that there may, in fact, be multiple valid answers to this problem. “No, there aren’t. If you knew the basic rules of mathematics, you would know that answer can only be 15. Multiplication ALWAYS precedes addition. Period!” Math problems aren’t up for interpretation. Period. This isn’t art class.

But what if addition were to precede multiplication? For centuries mathematicians have bent the rules that were handed down to them to explore new worlds like complex numbers, non-Euclidean geometry, and fuzzy logic, but this is not for you to do. Mathematics is not about experimentation or asking questions. Definitions are to be copied and memorized, not negotiated, and rules are meant to be followed and enforced (IN ALL CAPS!!!, if necessary).

Myth 4. Being smart means solving problems quickly.
Did you set a timer when you started this problem? This commenter did: “This took me 15 seconds. If it took u longer, you have issues.”

If Mad Minute exercises in school have taught us anything, it’s that math is not about careful, slow, and reasoned deliberation. It’s about being fast—shooting your hand up in the air before all your classmates and being the first to loudly drop your pencil when you finish an exam. Indeed, the prelude to this problem warned us to “answer fast if you are a genius.”

Should you need to count on your fingers or scribble on a napkin, or if you wish to take some time to play and to explore, we regret to inform you that you are, certifiably, not a genius. Or, as another commenter declared, “if you don’t get this within seconds you’re a mathsmuppet—FACT.”

Myth 5. Math is not for you.
It shouldn’t be surprising that this problem took you so long to answer or that you got it wrong. After all, the puzzle warns us that “98 per cent fails.”

You do not question this unreasonably high statistic, because math is clearly an enterprise for prodigies and savants. Math is for Albert Einstein. Math is for Matt Damon in Good Will Hunting and Russell Crowe in A Beautiful Mind. It certainly is not for you (and good riddance to it!). The headlines for other viral problems confirm your belief that math is a cold, inhuman instrument for categorizing and weeding out: “This maths riddle is baffling the internet . . . and only truly smart people are getting it right” (http://bit.ly/SunMath).

Treating the Infection

These math puzzles and the comments that accompany them have hidden and not-so-hidden messages that distort a field that is, at its core, deeply playful and creative. Unfortunately, school mathematics often reinforces these misconceptions, providing students with few opportunities to play with ideas, question assumptions, and explore possibilities.

“If I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making. . . . I couldn’t possibly do as good a job as is currently being done. . . . What a sad way to learn mathematics: to be a trained chimpanzee,” wrote Paul Lockhart in A Mathematician’s Lament (Bellevue Literary Press, 2002).

Hundreds of nearly identical drills later, a student may have memorized BEDMAS, but may have also learned to despise a beautiful subject that is the key to success in a variety of pursuits.

What kind of education might foster understanding, creativity, and appreciation of mathematics is a larger issue for another time. For now, a suggestion: Should a friend tag you on a viral problem because you are his token math person, politely disengage. Instead, share a link to Steven Strogatz’s series in the New York Times, to Evelyn Lamb’s blog in Scientific American, or to the work of another mathematician who shares the joy of his or her vocation. Maybe that mathematician is you.

Let’s shift the discourse around mathematics and reveal the beauty that many people have not had the privilege to see. And, if you tell two friends, who tell two more friends, who tell two more . . . the right message may just go viral.

Ilona Vashchyshyn, a high school math teacher in Saskatoon, Saskatchewan, maintains that the dress is white and gold.

Egan Chernoff, associate professor of mathematics education at the University of Saskatchewan, insists that the dress is blue and black.

Wednesday, March 29, 2017

Don’t Skip the Code

By Nathan Carter

I’ll just say it right out. You should learn to program.

Like, really learn it. At least a computer science minor, maybe more.

You’ve probably heard “coding is the new literacy” a thousand times. I’m not going to say that. My reason is different.

Are you expecting to hear about marketability and median starting salary for graduates who combine math and computer science? Sure, data science is hot, and you can make good money (http://bit.ly/RiseOfDataSci). But that argument is convincing only if making money is your goal. “I like topology!” you say. “Keep your stupid money!” Don’t worry—I’m not going to talk about your marketability.

So what is my reason? Is it those mathematicians who leverage computers in their research? The Borwein brothers are well known for their contributions to number theory from a computational approach. William Stein is at least as well known for creating the mathematical software package Sage as for his number theoretic uses of it. The four-color theorem was proven by a computer . . . sort of.

You’re getting warmer. After all, the first research team I joined after graduate school had me writing R code to analyze graphs. But no, that wasn’t my argument either; not every area of mathematical research needs computers.

Nor was I planning to use a close cousin of that argument, for educators. Programming gives math teachers the ability to create great interactive experiences like those on GeoGebra.org, the Wolfram Demonstrations Project, and ShinyApps.io. Great resources, but they’re not my reason why you should learn to program.

Give up yet? Okay, I’ll tell you.

You should learn to program because mathematics and computer science have an amazing synergy that will open up your brain.

No, not like head trauma. Like enlightenment.

Both mathematical language and programming languages are ways of making ideas precise. They reinforce one another, like weight training and sports. To learn to program well (not just quick and dirty coding but real programming) requires pervasive organization in your thinking. That organization transfers to how you think and communicate about everything, especially precise things like mathematics.

Heck, programming requires even more organization and precision than mathematics does! I’ll back that up with a quote from Deb Roy, a roboticist at MIT, who said, “To understand how something works, you need to build it.” I can’t tell you how many times I’ve found that to be true.

I didn’t understand the change of basis matrices in an abstract algebra course until I needed them in a C++ library I was writing for computer graphics. My computer has been more exacting on my handling of edge cases than any journal referee. Just this morning I learned that I didn’t fully understand a graph algorithm when I tried to implement it in a new context. In each case, my understanding had to improve because I wasn’t just doing math, but building it. There’s a stereotype that the hardcore geeks skip computer science to get to math. Don’t believe stereotypes.

I promised not to call coding the new literacy. It’s a common phrase, but I prefer the idea that modeling is actually the new literacy (http://bit.ly/NewLit).

Don’t run away, pure mathematicians! Yes, modeling happens in stats and applied math. But any time we turn ideas into a precise formulation we can explore, we’re modeling. Taking an idea you have in set theory, formulating it as a new axiom, and exploring its consequences with theorems is modeling too.

Mathematicians and computer scientists both model for a living. (Ha.) Learning to program will expand the ways in which your brain does modeling. That will make you better at it, in code and in math. That’s my real argument. You get a more awesome brain. It’s probably not too late to change your schedule for next semester.

Nathan Carter programs and does math. He teaches at Bentley University and has written two books for the MAA, Visual Group Theory (2009) and Introduction to the Mathematics of Computer Graphics (2016).

Friday, February 3, 2017

The Gods Have Descended

By Marc Chamberland

 Marc Chamberland
Some of the most deeply felt moments in life concern our connection to beauty— heart-stirring music, a baby’s laugh, a magnificent sunset. This same wonder applies to scenarios that would more often be described as technical rather than beautiful: the stunning precision of a dance troupe, a clever chess maneuver, a climber’s brilliant combination of moves on a rock face. What makes this techni- cal beauty so appealing? Perhaps it’s the uncom- mon mastery of a skill or the element of surprise. Beyond our analysis, however, these experiences capture our imagination and inspire our creative spirits.

But why do we seldom hear such stories connected to math? Defenders of mathematics can argue fervently about the mesmerizing beauty of their discipline, but it seems that their epiphanies are hidden and rarely celebrated. The mathematical community—indeed, the general public—could benefit from our tales of math- ematical allure. So, it’s time that I offer one of my own stories.

A Mysterious Series

In my first semester as an undergraduate at the University of Waterloo, Canada, I took Advanced Honours Algebra from Peter Hoffman. His tall, wiry frame, bulbous eyes, and 1970s shaggy hair came alive as he danced across the lecture platform. The University of Waterloo in Canada is a magnet for mathematical aspirants, so this 60-strong class was packed with some very bright students. One day Hoffman scribbled the following formula on the board:

“We’ve all seen this formula before, right?” he queried.

The only infinite series I had previously encountered was also the most accessible one: the geometric series. Although Hoffman’s equation is a standard result taught to calculus students, it was new to me. And it left me in awe. How could adding infinitely many polynomials—a mess in my mind—equal such a concise and elementary transcendental function? And where did the factorials come from? Aren’t those related to counting problems? My next question left me even more perplexed: How could somebody prove that this equation was true? To claim that such a formula was legitimate suggested madness, but to have a proof seemed divine.

I found all this so astonishing that a biblical phrase came to mind. The response of the people to seeing the apostle Paul perform a miracle in Lystra was my thought at seeing the new formula: “The gods have descended among us in the form of men.”

Many people, when overwhelmed by a stunning or surprising occurrence, experience a momentary shut- down of their chattering minds. It’s as if their brains need all available resources to process the experience. My reaction was a spontaneous response, an uncon- scious attempt to make sense of this inexplicably beautiful formula. Today, I routinely teach the mortal underpinnings of Hoffman’s formula, but it has never lost its wonder.

Unfortunately, mathematics is often taught as a col- lection of symbol-manipulating rules that are neither inspiring nor obviously applicable. Any good teacher knows that she will win over more hearts—and ac- companying good will—if she can show the wonder of her subject. Mathematics has much to teach students concerning beauty, usefulness, and connections to other disciplines.

Even if my students do not go on to do something groundbreaking with the math they learn, I hope that most of them will grow in their respect for, and even be charmed by, mathematical ideas. And if they are ever so awestruck, so captivated, so overwhelmed that their response to a new idea is something like “the gods have descended,” then I’ve succeeded in showing them that soul-stirring beauty can be found in mathematics.

Marc Chamberland is a professor of mathematics at Grinnell College and creator of the YouTube channel Tipping Point Math.