## 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.

## Tuesday, October 18, 2016

### I Love Teaching Math; Maybe You Will Too

By Patrick Honner

 Patrick Honner, assembling a dodecahedral sculpture.
Mathematics is a beautiful subject, full of compelling intellectual challenges and deep connections to virtually every aspect of life. And students of mathematics are rewarded with a wealth of career opportunities spanning science, engineering, technology, and the humanities. Coveted jobs in fields like statistics, computer science, and finance attract the mathematically minded, and for good reasons.

But one field doesn’t attract as many math students as it should: teaching. There are reasons for this, too— teaching doesn’t offer the pay, status, and opportunity that other mathematical jobs do. As a career, math teacher doesn’t seem like an optimal solution.

Yet, it should. Because being a math teacher offers its own great rewards. Of course, I’m biased. I’ve been teaching high school math for nearly 15 years, and I love it.

But I didn’t start out wanting to be a teacher. After college I went to graduate school, lived abroad, and worked for a variety of tech companies. I enjoyed the options that studying math had given me, but I didn’t find my place right away. Becoming a teacher was a natural decision for me. I loved math and enjoyed teaching—as a tutor, as a teaching assistant in graduate school, and as an English teacher abroad. But it took a few years to realize what a great job it was.

First, being a math teacher is a wonderful mathematical challenge. Each student sees math from a unique perspective, which is often very different from my own. Finding ways to make our mathematics meet requires me to understand ideas in multiple ways, which is one of the most powerful and exciting aspects of mathematics. And it’s something I get to do, and learn from, every day as a teacher.

Teaching math also requires more creativity than I imagined. The need for new ways to introduce ideas, connect concepts, and engage students inspires me to innovate to create compelling problems, tasks, and projects at the right level of complexity.

And teaching has inspired me to be more creative with mathematics. I photograph the math around me, write about my mathematical experiences, and build using mathematical tools. It is personally fulfilling, but it also inspires my students, who in turn inspire me with their geometric photography, algorithmic art, 3D sculptures, and mathematical writing.

Through teaching, I have grown as a mathematician. I have to develop multiple conceptions of mathematical ideas, distill complex systems and procedures to their essence, and identify and highlight the fundamental principles that unite disparate, disconnected curricula. I have a much deeper understanding of mathematics because of all this.

And of course, the work is profoundly meaningful. As a teacher, I never wonder if what I do makes a difference. Every day I help students move forward in their lives—through understanding mathematics, the world, and themselves. I know what I do has an impact. I feel it every day: when students share their own mathematical experiences with me, when graduates tell me they want to study math in college, and when former students tell me about how they are applying math in their careers.

Teaching can be a great job. But it’s not an easy job. Under the best circumstances, teaching taxes your intellect, tests your emotional resolve, and humbles you. And few teachers work under the best circumstances. It’s not for everyone. But it is a job where, after 15 years, you can feel as energized and passionate as when you started, where you know you can continue to grow and evolve, and where you know you make a difference.

The next time you think about math teaching, think about what a great job it can be. Maybe it’s not the right job for you right now, but you never know. Maybe, like me, you might find your life’s optimal solution.

Patrick Honner teaches at Brooklyn Technical High School. He’s a three-time Math for America Master Teacher and a recipient of the Presidential Award for Excellence in Mathematics and Science Teaching. He blogs at MrHonner.com and is @MrHonner on Twitter.

## Wednesday, August 31, 2016

### A Mathematician Goes to Washington

By Katherine Crowley

 Senator Al Franken and Katherine Crowley. Photo Credit: Katherine Crowley
In December 2015, President Obama signed into law a replacement for the much-maligned No Child Left Behind Act. The new law addresses a wide range of education issues, from curriculum to testing to teacher evaluation. In the section on testing requirements, a new provision allows states to choose computer-adaptive—rather than pencil-and-paper—tests to meet federal testing requirements.

Five years earlier, as an American Mathematical Society congressional fellow, I was putting this new testing bill together for my boss, Senator Al Franken. It was supposed to be easy. The bill had been requested by principals, teachers, and parents, promised shorter test times and higher quality feedback, and cost nothing. I had worked on more controversial issues for the senator and felt lucky this time to have such a clear path forward. My relief was naive. If not for my experience as a mathematics professor, I might not have succeeded.

When trying to pass a bill on Capitol Hill, the most productive thing you can do is anticipate who will oppose you. Then call them, and listen.

That is how I learned that advocates for students with disabilities, who have worked fiercely for decades to ensure equal access to education for these students, had legitimate concerns. They wondered whether these computer-adaptive tests—which, like the GRE, ask harder questions if you answer correctly, and easier questions if you answer incorrectly—might unfairly characterize students with less common learning paths. The senator did not want to proceed without the support of these advocates, and their opposition was firm.

Luckily, when you work for a U.S. senator, everyone calls you back. So I was able to talk to the mathematicians designing the computer tests and learn the exact algorithms that determine how students’ abilities are measured. Then, drawing on years of teaching experience, I translated that information for the advocates. The advocates endorsed the bill and even adopted computer-adaptive testing as a top policy goal.

Most days on Capitol Hill, I didn’t use mathematics; I did policy. I worked on student loan reform, which eliminated federal subsidies to banks and saved taxpayers \$60 billion. I worked to ensure access to school meals for America’s poorest kids during the height of the recession. I secured cosponsors for the Student Non-Discrimination Act, which would add civil rights protections against discrimination in schools for LGBT students.

These projects and others meant preparing the senator for meetings, advising him on votes, building support for his legislative ideas, and crafting strategies to pass these ideas into law. It was incredibly exciting.

Why a Mathematician?

But what is the point of having a mathematician do this? One reason is that the mathematics community has a lot at stake in national policy. Congress will write STEM bills whether there are scientists and mathematicians in the room or not. Letters for mathematics and science funding can easily go unnoticed; I was in a position to make sure they crossed the senator’s desk. When the senator’s support for one bill hinged on understanding the science behind it, I tapped into my network of science policy fellows to find that expertise. There are critical moments when it really matters that we, as a discipline, are there.

The best reason to work in policy as a mathematician is because it appeals to you. Whether in service of our discipline or our country, you will negotiate agreements that improve the lives of millions of Americans. The mathematics in which you’ve invested so much time and passion will play a role, sometimes directly, but often indirectly, because you will have to be clever. You’ll strategize, you’ll get cornered, you’ll have your arms twisted. Sometimes, you’ll succeed in outsmarting everyone anyway. To preserve a strong mathematics community, we need good mathematics, and we need good policy that supports it. Choose the path that inspires you, knowing how critical both pieces are.

Katherine Crowley worked in policy in Washington, D.C., both on Capitol Hill and at the Department of Energy. Her PhD in mathematics is from Rice University.
Email: katherine.d.crowley@gmail.com

## Thursday, March 31, 2016

### Embrace Mistakes

By Eduardo Briceño

Mistakes can be frustrating, embarrassing, and disheartening, which may lead us to always want to avoid them. But avoiding mistakes keeps us from taking on challenges we can learn from.

If we’re inside Hermes, the spacecraft in the book and film The Martian, calculating the velocity needed to intersect Mark Watney and bring him home, we need to get the right answer or he’ll die. Ideally, we want an expert astrodynamicist to work on that problem using skills she has already mastered so that she doesn’t make an error. But if that astrodynamicist had never challenged herself to tackle problems beyond what she knew, inevitably making errors and learning from them along the way, she never would have built the expertise needed to become an elite member of NASA.

Mistakes to Avoid or Pursue

Distinguishing mistakes we want to pursue from those we want to avoid helps us learn more effectively.

If we desire a high rate of improvement, we must pursue stretch mistakes. These happen when we work on skills we haven’t mastered. If we’re attempting a task that we don’t know how to do yet, we’re bound to make mistakes. When we try to solve a tough math problem, we can learn a lot by reviewing our work and identifying where we made mistakes and what we can learn from them. We pursue stretch mistakes not by trying to do things incorrectly, but by attempting tasks that are challenging. Thereby we learn and grow.

The aha-moment mistake happens when we do something as we intended, but then realize it was a mistake to do so. For example, if our astrodynamicist- in-training is trying to predict the trajectory of an object moving very fast, she may apply Newton’s laws of motion and then realize that the object is moving in ways not predicted by the model. That may lead her to discover that she must take into account Einstein’s special relativity. Although we can’t seek out aha- moment mistakes, when they happen, we can treasure them as learning opportunities.

Some mistakes are not as desirable. We want to avoid sloppy mistakes, which are errors we make when doing something we have already mastered. But we’re human and sometimes make them. When we do, let’s learn from them. We can examine what led to the error and decide how we could change our processes to avoid them. Perhaps it was a lack of focus—sloppy mistakes can be good reminders to minimize distractions, slow down, and pay attention to the details.

Finally, we can minimize high-stakes mistakes— mistakes that could have disastrous consequences. A high-stakes situation could be one in which lives are at risk, such as when saving Mark Watney or designing a bridge.

Non-life-threatening situations may also be consid- ered high stakes, such as a college entrance exam or job interview. In those situations, we may focus on what we have mastered rather than on what we’re learning. After we perform, whether successful or not, we can reflect on what we can learn from those experiences, back to seeking out new challenges.

Some teachers (and our grade-conscious education system) may inadvertently send the message that mis- takes are undesirable. But learners who don’t take the difficult classes and who don’t try the challenging problems miss the opportunities to make mistakes, analyze the thinking that led to them, learn from such confusion, and improve. We learn the most when we view mistakes as opportunities to enhance our abilities.

So what challenge will you tackle next, and what will you do when you make your next mistake?

Eduardo Briceño is the cofounder and CEO of Mindset Works. He and his colleagues write regularly at mindsetworks.com.

## Wednesday, February 3, 2016

### The Law of the Broken Futon

By Ben Orlin

Try asking random adults about their math education. They refer to it like some sort of NCAA tournament. Everybody gets eliminated, and it's only a question of how long you can stay in the game. "I couldn't handle algebra” signifies a first-round knockout. “I stopped at multivariable calculus” means “Hey, I didn’t win, but I’m proud of making it to the final four.”

It’s as if each of us has a mathematical ceiling, a cognitive breaking point, beyond which we can never advance.

But there’s a new orthodoxy among teachers, an accepted wisdom that just about anyone can learn just about anything. It takes grit, effort, and good instruction. But eventually, you can bust through any ceiling.

I love that optimism, that populism. But if there’s no such thing as ceilings, then what do students keep thudding their heads against?

Is there any way to bridge this canyon-wide gap in views?

I believe there is: the Law of the Broken Futon.

In college, my roommates and I bought a lightly used futon. Carrying it up the stairs, we heard a crack. A little metallic bar had snapped off. The futon seemed fine—we couldn’t even tell where the piece had come from—so we simply shrugged it off.

After a week, the futon had begun to sag. “Did it always look like this?” we wondered.

A month later, it was embarrassingly droopy. Its curvature dumped all sitters into one central pig-pile.

And by the end of the semester, it had collapsed in a heap on the dorm room floor.

Now, Ikea furniture is the fruit fly of the living room: notoriously short-lived. There was undoubtedly a ceiling on our futon’s lifespan, perhaps three or four years. But this one survived barely eight months.

In hindsight, it’s obvious that the broken piece was crucial. The futon seemed fine without it. But day by day, butt by butt, weight pressed down on structures never meant to bear the load alone. The framework warped. The futon’s internal clock was silently ticking down toward an inevitable failure.

And, sadly, so it is in math class.

Say you’re acing eighth grade. You can graph lines, compute slopes, specify points. But if you’re missing one vital understanding—that these graphs are the x-y pairs satisfying the equation— then you’re a broken futon.You’re missing a piece upon which future learning will crucially depend. Quadratics will haunt you; the sine curve will never make sense; and you’ll bail after calculus, consoling yourself, “Well, at least my ceiling was higher than some.”

Why not wait to add the missing piece later, when it’s actually needed? Because that’s much harder. In the intervening years, you develop shortcuts that do the job, but warp the frame. You’ll need to unlearn these workarounds—bending the futon back into its original shape—before you can proceed.

Once under way, damage is hellishly difficult to undo.

This, I believe, is the ceiling so many students experience in high school and early college. It’s not some inherent limitation of their neurology. It’s something we create. We create it by prizing right answers over deep reasoning. We create it by saying, “Only clever people will get it; everyone else just needs to be able to do it.” We create it by saying, in word or in deed, “It’s OK not to understand. Just follow these steps and check your answer in the back.”

We may succeed in getting the futon up the stairs. But something is lost in the process. Moving forward without key understandings is like marching into battle without replacement ammo. You may fire off a few rounds, but by the time you realize something is missing, it’ll be too late to recover.

A student who can answer questions without understanding them is a student with an expiration date.

Ben Orlin is a teacher in Birmingham, England. His blog is Math with Bad Drawings.
Email: ben.orlin@gmail.com