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

Wednesday, April 1, 2015

History Helps Math Make Sense

By Daniel E. Otero

Has it ever bothered you that many mathematics textbooks begin with a number of strangely crafted definitions?

These definitions invariably turn out to be extremely valuable for the development of the theory in question, but it may be days, months, or years—with many rereadings—before this is apparent.

I remember wondering as a college sophomore why the definition of the linear independence of vectors v1, v2, …, vn had to be so complicated and, in particular, what the algebraic condition

The only scalars a1, a2, …, an that satisfy a1v1 + a2v2 + … + anvn = 0 are a1 = a2 = … = an = 0.

had to do with the ability of the vectors to fill out n-dimensional space. It was years before I figured out that connection.

Anyway, after the cryptic definitions, the textbook author embarks on proving a series of theorems whose purpose is hidden until quite late in the theory’s development, if ever.

The most important results, so identified because they are called The Fundamental Theorem of Something or Other, appear at the end of section 3.3 as a corollary to some other impenetrably technical theorem, apparently as an afterthought!

If the intrepid reader has lasted this far, the author throws a bone late in chapter 4 in the guise of an application of the theory to some problem that may have helped someone at some time.
No wonder so many people think that mathematics is only for nonhumans!

To be fair, more and more mathematics textbooks are far better written than this caricature I have painted for you, but sadly, plenty of examples of expositional writing in mathematics fit this mold.

You can thank Euclid for this penchant professional mathematicians have of organizing their writing in axiomatic form: definitions, axioms, propositions, theorems, and corollaries. Indeed, Euclid didn’t bother adding applications to such expositions. You can thank Archimedes, Ptolemy, and Galileo for including them (although Descartes, Gauss, and Cauchy usually did not).

I won’t argue that the traditional axiomatic style lacks value—mostly because I don’t believe this at all! I will suggest that it is not the best vehicle for learning mathematics. My contention here is that the best antidote for students who struggle with traditional forms of axiomatic exposition is to investigate the history of the subject.

That definition of linear independence at the start of this article? I finally started figuring out the link between it and the geometry of space when I read about linear algebra’s history (specifically the work of Herman Grassmann in the mid-1800s and the later formalism of linear algebra under Giuseppe Peano and others later in that century). And this is not the only occasion when learning how a mathematical subject developed helped me make sense of what was going on.

Studying the history of mathematics has much to offer the mathematics student:
  • Context (the conceptual and cultural circumstances for the underlying problems); 
  • Motivation (the rationale or even the value of wanting to know the answers to the central questions involved); and 
  • Connections (how the mathematicians thought in terms of other ideas that were already established at those times and places). 
 What’s more, the history of mathematics humanizes the subject in a way that no formal presentation can, reminding us that it’s people who do mathematics, not textbooks. And that, after all, just makes sense.

To purchase at JSTOR: http://dx.doi.org/10.4169/mathhorizons.22.4.34

Danny Otero is an associate professor of mathematics at Xavier University, president of the MAA Ohio Section, and chair of the History of Mathematics Special Interest Group of the MAA. He still digs the Power Puff Girls. Email: otero@xavier.edu

Sunday, February 1, 2015

I Love Math And I Hate The Fields Medal

By Cathy O'Neill

I’ve loved math for as long as I can remember. When I was five I played with Spirographs and learned about prime numbers, and in high school I solved the Rubik’s cube with group theory. Gorgeous stuff! Inspiring!

In college, I was privileged to learn algebra (and later, Galois theory) from Ken Ribet, who became my friend. He brought me to dinner with all sorts of amazing mathematicians: Serge Lang, J. P. Serre, Barry Mazur, John Tate, his Berkeley colleagues Hendrik Lenstra and Robert Coleman, of course, and many others.

Many of the characters behind the story of solving Fermat’s Last Theorem were people I had met at dinner parties, including Ken himself. Math was discussed in between slices of Cheese Board Pizza and fresh salad mixes from the Berkeley Bowl.

The best thing about these wonderful people was how joyful they were about the serious business of doing math. It was a pleasure to them, and it made them smile and even appear wistful if I’d mention my difficulties with tensor products, say. They were incredibly generous to me, and honestly I was spoiled. I had been invited into this society because I loved math and was devoting myself to it, and that was enough for them. Math is, after all, not an individual act; it is a community effort, and progress is to be celebrated and adored. And it wasn’t just any community. It was an exceptionally nice group of people who loved what they did for a living and wanted other cool, smart people to join them.

I mention all this because I want to clarify that in such a community, where math is so revered and celebrated, it is its own reward to prove a theorem and tell your friends about it.

Now that I’ve explained how much I love math, let me explain why I hate the Fields Medal. Through the filter of that award, the group effort I’ve just described is utterly lost, is replaced with a synthetic and false myth of the individual genius working in isolation.

You see, journalism has rules about writing stories that don’t work for math. When journalists are told to “put a face on the story,” they end up with all face and no story. After all, how else is a journalist going to write about progress in some esoteric field?

The mathematics is naturally not within arm’s reach: It is by nature deep and uses multiple layers of metaphor and notation that even trained mathematicians grapple with, never mind a journalist, and never mind a new result on the far edge of what is known. Too often the story becomes about what the mathematician had for breakfast the day of his or her discovery rather than what the discovery itself means.

The Fields Medal, which is easy to understand (“it’s the Nobel Prize for math!”), is thus incredibly and dangerously misleading. It gives the impression that we have these superstars who “have it” and then we have a bunch of wandering nerds who “don’t really have it.” That stereotype is a bad advertisement for mathematics and for mathematicians. Plus, the 40-year-old age limit for the award is just terrible and obviously works against certain people, especially women or men who take parenting seriously. And while the fact that a woman has won the Fields Medal is a good thing, it’s a silver lining on in otherwise big old rain cloud, which I do my best to personally blow away.

Lest I seem somehow mean to the Fields Medal winners, of course they are great mathematicians, all of them. To be sure, there are many other great mathematicians who never get awards, and awards tend to be given to people who already have a lot of resources and don’t need more. Even so, I’m not saying they shouldn’t be celebrated, because they’re awesome, no question about it.

I’m just asking for more celebrations. I would love to see some serious outward-facing science journalism celebrating the incredible collaborative effort that is modern mathematics.


Cathy O’Neil is a mathematician and a data nerd. She wrote Doing Data Science and is working on a book called Weapons of Math Destruction. She writes regularly at mathbabe.org.

Email: cathy.oneil@gmail.com