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