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.
 Twitter: @ebriceno8

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

Tuesday, November 3, 2015

Start with Art

By Eve Torrence

Eve Torrence holding her sculpture Blizzard Three.
Photo Courtesy of Randolph-Macon College.
If music and math have so much in common, why do so many people love music but hate math? This is a paradox that mathematics should exploit as a way to improve our image with the public. Dare we have the courage to be loved? Schoolchildren study both math and music. Both involve learning abstract notation and practicing applying that notation. Both require hard work for success. But music is often a special weekly treat that children are excited to explore, while math can become daily rudgery. This is, of course, not universally true for every student or every classroom. And I do not mean to diminish the great progress that has been made in mathematics education. Yet we could do much more to improve the image of mathematics as beautiful and worthy of enjoyment.

Music can be appreciated on many levels. Even infants seem to enjoy music. Why not mathematics? Perhaps our approach is wrong. We don’t make children learn music composition before they ever hear a tune. Why can’t we introduce students to a beautiful piece of mathematics at a young age?

The growth in the field of mathematical art may be a step in the right direction. Here is a way to express the beauty of mathematics in a way that is accessible to everyone. It is the perfect visual balance to music’s auditory appeal. With modern technology, we can produce images and objects that demonstrate complex concepts in ways that had been impossible. If you have never seen the Exhibit of Mathematical Art at the Joint Mathematics Meetings, your mathematical education is incomplete. Luckily the computer age not only makes many of these pieces possible, but it also makes them possible to view at bridgesmathart.org/bridges-galleries/art-exhibits.

Mathematicians know that mathematics is a creative subject, but other people laugh at this idea. How can there possibly be creativity in mathematics? Most people are never exposed to this concept, let alone see a demonstration of it. Yet everyone knows that music is a creative endeavor, even if he or she has never studied an instrument and cannot read music. The time has come to open up access for all to the creative world of mathematics.

Not everyone has the opportunity or desire to learn about hyperbolic geometry, but everyone can enjoy Start with Art AFTERMATH Escher’s Circle Limit prints. We should accept this as legitimate math appreciation. The difficulty of our subject is not a valid excuse for our sometimes-elitist attitude toward its enjoyment. Those who want to explore the concepts exhibited in a piece can delve into the topic as deeply as they wish. Not everyone needs to understand music theory to appreciate Bach. Not everyone needs to understand circle packing to appreciate what Robert Lang can accomplish in origami (see http://bit.ly/1TWQrQK).

But anyone can appreciate that Lang’s work is an extraordinary accomplishment made possible through his knowledge and application of mathematics. And seeing Lang’s work has drawn many students into studying the mathematics behind his art. How many students might be drawn to studying mathematics if we could change the way they think of our subject?

We need to expose the public to the fact that mathematics is not simply arithmetic and polynomial factorization. The arts are a way to shed light on the diversity, creativity, and progress of modern mathematics. Every schoolchild should have the chance to see mathematical art. Perhaps someday mathematics, like music, will be thought of as a subject for lifelong interest and enjoyment. It is never too late to learn to play the piano—or study field theory.

Eve Torrence is a professor of mathematics at Randolph-Macon College and past president of Pi Mu Epsilon. She loves the symmetric beauty of polyhedra and sharing mathematics through her sculptures. Email: etorrenc@rmc.edu

Thursday, September 3, 2015

The Common Core for Mathematics in a Nutshell

By Christopher Danielson

If everything you, like many Americans, know about Common Core State Standards (CCSS) comes from social media, you likely think children are being taught to subtract by adding, to write letters to Jack instead of learning their facts, and to draw endless series of dots. (If you got none of those refer- ences, a web search will quickly bring you up to speed.)

As with most urban legends, the commonly voiced concerns about the CCSS begin with a kernel of truth and spin into fiction and fear. Here are the things you, as a mathematically concerned citizen, need to know about these standards.

For U.S. elementary students, addition and subtraction have been taught as distant cousins. The facts are learned separately; the algorithms require substantially different (and seemingly unrelated) procedures; each has its own set of keywords that appear in word problems.

A major goal of the CCSS writing team was coherence. To view operations coherently means to study their interrelationships from the earliest stages. Viewing addition and subtraction as related operations sets the stage for operating on integers in middle school (One interpretation of is “What do I add to to get 7?”). It gives students tools for devising efficient computation strategies (Do you really want to use the subtraction algorithm to solve And it sets students up to more fluently solve algebraic equations in ways that make sense.

The Standards for Mathematical Practice are an important feature of the CCSS, and all eight standards apply to all grade levels. (Go read them—you’ll find them useful for thinking about college-level math as well!)

Perhaps my favorite of these practices is: “Construct viable arguments and critique the reasoning of others.” In the early grades, this might mean saying how you know that eight empty pistachio shells means you ate four pistachios, not eight. In middle school, it might mean explaining how dividing by a fraction can yield a quotient larger than the original value. In high school, it might mean writing a geometric proof.

Students in Common Core classrooms have a lot of practice arguing the truth of their mathematical ideas,establishing importantexpectations about what it means to do mathematics, and easing their transition into formal proof writing in geometry and in college mathematics.

Another Student asks students to “look for and make use of structure.” As an example, the distributive property is a structure that underlies basic arithmetic, algebra, calculus, and more. It explains why multidigit multiplication algorithms work, why some (but only some) quadratic expressions factor nicely, and why the definite integral of a constant multiple of a function equals that same multiple of the integral of the function.

Looking at the distributive property as a connecting structure across many contexts brings coherence to what could otherwise be seen as a long list of disconnected, hard-to-remember facts. In contrast to a common mnemonic device for multiplying binomials (FOIL, anyone?) the distributive property applies broadly and provides a foundation for making sense of various mathematical situations. CCSS pushes teachers, curriculum writers, and students to focus more effort on these larger structural ideas and less effort on memorizing individual instances of them.

Going Deeper

If you are interested in the K-12 math educational system, you’ll want to know more than what I’ve covered here. The CCSS for Mathematics, which are neither overly technical nor lengthy, are available at corestandards.org. A series of “progressions documents” expands on the standards with research references and text that helps the reader see how understanding builds across grade levels (ime.math.arizona.edu/progressions).

I hope that this information and my book, Common Core Math For Parents For Dummies, provide an antidote to the misinformation so easily encountered online, and that they spur readers to become better informed and better equipped to formulate viable arguments about mathematics teaching practice.

To purchase at JSTOR: Math Horizons

Christopher Danielson teaches and writes in the Twin Cities of Minnesota. He is no dummy, nor does he believe his readers to be. Email: mathematics.csd@gmail.com