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

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

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, …, v_n had to be so complicated and, in particular, what the algebraic condition

The only scalars a₁, a₂, …, a_n that satisfy a₁v₁ + a₂v₂ + … + a_nv_n = 0 are a₁ = a₂ = … = a_n = 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

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.

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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