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 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 (

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:

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:

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:

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


Tuesday, November 11, 2014

When Will I Use This?

Douglas Corey—Brigham Young University

The top search engine completions for "when will I use . . ." are all related to school mathematics. Some students ask such questions as a challenge to the teacher, but others sincerely want to know. Like me, you may have a hard time remembering the last time you used multiplication for something other than schoolwork, yet we use multiplication all the time. It is so ingrained in our thought processes that we don't notice it.

When someone asked when I had last used it, I eventually remembered that it was to calculate the area of my raspberry patch. But this example probably wouldn't convince a skeptical student that math is useful. Examples about baseball statistics, recipe conversions, or grocery store price comparisons may also be unconvincing. Why? Because if an application is outside the interest of a student, he or she discounts it. In truth, a teacher doesn't know when a student will use the math being taught (except on the exam). It is fraudulent to pretend otherwise.

Connecting the Dots

Typically we don't know what we don't know. This makes it very difficult to predict what kind of knowledge we will need. It is also very hard to see how we could use knowledge that we don't have.

In Steve Jobs's commencement address at Stanford University, he describes taking a calligraphy class in college. In the class he learned about typography: the technical aspects, the terminology, its history, and the characteristics of a beautiful typeface. This class, which at the time had no practical applications for him, led to his inclusion of multiple typefaces and proportionally spaced fonts in the first Macintosh computer. Jobs said, "You can't connect the dots looking forward; you can only connect them looking backwards. So you have to trust that the dots will somehow connect in your future."

The Eye of the Mind

Knowledge enables us to see what others can't. When one of my sons puts his shirt on inside out and backwards, I think of the symmetry group generated by the actions on his shirt. When I watch the balls on my kids' trampoline roll around, I think about how their paths are modeled by hyperbolic geometry and how the model also governs the path of light through space.

It doesn't go the other way. People don't stand on the trampoline and ask themselves what connection it has to hyperbolic geometry or to Einstein's theory of general relativity. They don't think about abstract algebra when they see a shirt turned inside out. They can't see these connections, so the connections don't exist to them.

Just Look It Up

Students argue that it is a waste of time to memorize formulas, definitions, theorems, and proofs because they can always look up such things. But we look up things only when we know we don't know about them, and we need to know fairly specifically what we don't know in order to search for it.

As an experiment, find a meaningful quote, such as this favorite of mine attributed to Thomas Edison, "We often miss opportunity because it is dressed in overalls and looks like work." Repeat it every morning and evening for two weeks. During this period you'll find that the quote comes to mind as relevant multiple times. Without having memorized it, you would not have stopped and thought, "I wonder if there is a quote by Edison that I could use right now?" You did not know enough to see any connection to Edison's ideas.


No one knows when you will use the math you are learning in your classes. Most knowledge gets applied to situations we never anticipate. It pays to learn all you can about all you can. You will be able to see how you benefit from it only by connecting the dots backwards.

An expanded version of these arguments, which I give to my students, is available at

Douglas Corey is an associate professor in the mathematics education department at Brigham Young University. He stays busy with his eight kids, all of whom are girls but seven.

This article was published in the November 2014 issue of Math Horizons.

Monday, September 1, 2014

Bad at Math Is a Lie

Matt Waite—University of Nebraska-Lincoln

All I had to do was test into college algebra. That was what the College of Journalism and Mass Communications at the University of Nebraska required in 1993. If you could score well enough on the math placement exam for incoming freshmen to get into college algebra—not take it, just get into it—you were done with math.

I very nearly didn’t graduate from high school on time because of math, so this made the test a five-alarm panic attack. I struggled in every math class I took in high school. I needed tutors and small miracles to pass.

But all of this was OK because I was Bad at Math. It was a thing. People I knew were Bad at Math. My mom was too. So it was probably genetic. I was born to agonize over math, my friends weren’t.

So when I tested into college algebra, you would have thought I hit a home run in the bottom of the ninth in the seventh game of the World Series and scored the goal that won the World Cup all at the same time. I went running across campus jumping and pumping my fists like a lunatic.

Fast forward almost 20 years, and there I was, taking that same math placement exam. I wanted to get an MBA, and calculus was required to get in.

You could say I bombed, but that’s not true. I tested exactly where I should have: remedial algebra. I was going to have to take two math classes before taking calculus.

That’s how I—a 37-year-old father of two, a professor with a résumé that includes reporting from a war zone, stargazing through the eye of a hurricane, starting my own software company, and building a website that was the first to get a Pulitzer Prize—ended up in a remedial algebra class with students half my age.

And I was terrified.

See, I was Bad at Math. I knew I was going to have to sit in the front row, ask millions of questions, and work harder than anyone in there if I had any hope of passing. Forget about getting a good grade.

And that’s what I did. I sat in the front row. I raised my hand so much they asked me to stop. I was that student. The one who did extra homework. The one who started studying for tests a week ahead of time.

And I learned something: Bad at Math is a lie. It’s a lie I believed to make struggling at math hurt less.

I worked harder in that one math class than I had in whole years of schooling. And I got an A in math for the first time since the fifth grade. And I did the same thing in College Algebra—and got the same grade.

When it came time to take calculus, I was beyond scared. I struggled. I went in for extra help. I used online videos. I did twice as much extra homework. And I lay awake at night, worrying, going over problems, doing them in my head.

I might get the A in calculus tattooed somewhere. It means that much to me.

Somewhere in the early years, I missed something. I daydreamed through a lecture, something. Something didn’t click, I didn’t ask, and I started struggling. The Bad At Math lie was born. And I believed it.

But the truth is anyone can get math. Some of us just have to work harder. Some of us didn’t get the message that you have to practice. We didn’t get that math is really explaining how to solve a problem, not just solving the problem in front of you.

If you’re reading a magazine about math, none of this might make any sense to you because you get it. But trust me, someone sitting near you is in agony, panicking that he or she doesn’t get it.

Someone near you still believes the lie.

Matt Waite is a professor of practice at the College of Journalism and Mass Communications at the University of Nebraska–Lincoln, teaching reporting and digital media development. He was the principal developer of the website PolitiFact.

This article was published in the September 2014 issue of Math Horizons.

Wednesday, April 2, 2014

Every Math Major Should Take a Public-Speaking Course

Rachel Levy—Harvey Mudd College

In mathematics courses we practice deep thinking, clear writing, and effective problem solving. Excellent public speaking complements these skills. As one of my students put it:

No matter what we all do after college . . . [we] will have to speak to people. Every one of us will have a limited amount of time that we can convince someone else to see our point of view.

A public-speaking course can help you develop a superpower: the ability to communicate to a live audience in a clear, compelling manner. Every mathematics major should take such a course. Comments in italics are from my students in Math Forum, our required public speaking course at Harvey Mudd College.

Unfortunately, standing up in front of a group makes us nervous. Our hearts beat faster; this throws off our body chemistry and can make us feel ill. We fidget, rock back and forth, make awkward hand gestures, or stand unnaturally still. Our body language, voice inflection, and gestures reveal our discomfort.
There’s something fundamentally nerve-racking about giving a presentation. The first day of Math Forum, we all attempted to describe that near-indescribable feeling of speaking in front of an audience . . . Here I was, in a class that I had dreaded taking since hearing about it my freshman year, thinking I was the only person that had these feelings, . . . and yet the dozen other people in the class shared this same feeling.

Practice is key to taming our nervousness and to making a successful presentation.
With each talk I delivered throughout the semester, my confidence only increased. For my first talk, I was a nervous speaker that feared the whole ordeal, unable to deliver my opinions with sincere confidence. In contrast, for my last 10-minute talk, I was completely comfortable and calm. I had even begun to enjoy interacting with the audience during the presentation.

We can learn a lot from watching other speakers—professors, renowned lecturers, and classmates. If possible, watch yourself giving a presentation.
In re-watching the video of my second talk, . . . I noticed I sometimes shifted my body weight from one side to the other. . . . In my [later] talk, I felt at ease, and this was evident in my posture.

Careful preparation is essential to a first-rate lecture. Speaking tasks often have a fixed, typically short, time allotment. In a public-speaking course, you learn to deliver a message within a given time and to pare your talk down to its essence, so that there is no wasted moment. Although there are many ways to construct a successful presentation, you’ll learn how to write a strong introduction and conclusion, and how to connect them with a logical flow of ideas.

Written mathematics can be expressed elegantly and efficiently with words and symbols, but in a presentation, written words and complicated mathematical notation are difficult to follow. A lecture benefits from an effective use of images, clever uses of color, and careful placement of choice information for each slide. Generally: Less is more.

These are a small sampling of the many skills you will learn in public-speaking class. If no such class is available, consider other options, such as a theater class or a Toastmasters club; research groups and math clubs can also provide opportunities for you to give a presentation. These experiences can help you deliver confident, compelling communication about any topic, including mathematics.
I got a glimpse at the sorts of strategies I’ll need to give good talks: approaching a topic from the eyes of someone who is unfamiliar with it, eschewing notation unless it is particularly elucidating, leaving out ideas that don’t support whatever central message I want to present, and many more. . . . I am happy I was able to share the ideas I find interesting with the rest of my Forum class. It was unbelievably satisfying to finally give a talk I could be proud of.

Rachel Levy is an associate professor of mathematics at Harvey Mudd College and editor-in-chief of SIAM Undergraduate Research Online (SIURO).

This article was published in the April 2014 issue of Math Horizons.

Sunday, February 23, 2014

Steven Strogatz on Mathematics Education

Patrick Honner: What are your thoughts about the state of math education right now?

Steven Strogatz: My thoughts are mostly based on my own instincts as a teacher and what I’ve seen of teachers I admire. I don’t know much about the constraints that practicing teachers face in high schools right now, so my opinions are fairly uninformed. But I do worry about math communication and teaching in general.

Can I give you my “I have a dream” speech?

PH: By all means!

SS: In my dream world, everyone would have the chance to be a teacher the way Mr. Joffray [Strogatz’s high school calculus teacher and the subject of his book The Calculus of Friendship] was a teacher. His job was to teach us calculus, but he had his own vision of how to teach it and he followed that vision. He was creative, and he put his personal stamp on the course for us. He trusted his judgment, and the school trusted him. He could teach us the way he wanted to teach us, and he was a great teacher.

This is a profession that should be revered. What’s more important than teaching? Why not let teachers teach creatively and inventively? So that’s my dream: a world in which teachers are given the freedom to teach the subject they’re supposed to teach, the way that makes sense to them.

PH: You have two daughters in school right now. Do you think they are being exposed to math in a positive way?

SS: No, I don’t. I worry that my kids are not falling in love with math because it’s being presented as lots of procedures that they need to learn.

It’s too fast. My eighth-grade daughter is taking algebra, and one day she’s doing word problems, like “find three consecutive odd numbers that add up to 123,” and the next day she was doing something I’d never heard of—literal equations.

It just struck me as unbelievable that we’re doing word problems in one night’s homework. Students should spend at least two to three weeks on word problems. They’re hard! Every old-fashioned word problem is being thrown at her in one night.

PH: And then it’s off to literal equations the next day.

SS: I can’t imagine what any kid is doing who doesn’t have a math professor as a parent. The whole thing looks crazy to me. I’m sure even my daughter’s teacher doesn’t want to do it this way. Something is really messed up.

PH: Should math be a mandatory subject for kids?

SS: I’m conflicted about it—I don’t know what to think. There are a lot of students out there who would love math but don’t know that. So they have to be exposed, or maybe even forced, to take math to realize they like it. But after a certain amount of that, it becomes clear to a student that they don’t want to take more math. We as a profession should think about this again.

PH: What math do you think all people should know?

SS: Some amount of number sense is essential—for example, to know what it means when the store says certain items are 20 percent off. If you don’t know what that means, to me, you’re not educated. I feel comfortable saying that every person should understand fractions. But after that, what? Does a person need to know what a polynomial is? That’s not clear to me.

What should a person learn, if anything, after arithmetic? That seems like a pretty interesting pedagogical question, and I don’t believe our current curriculum is the optimal answer. Algebra I and II are good subjects, but so is network theory. It would be nice if people could understand how Google works, for example; it’s not that hard.

There’s a lot of fun in math. Do we really have to teach such dead material? If we could get a cadre of
people who love math and who get it the way you get it or the way I get it—people who know what math is about—you don’t need to tell them how to teach. You just leave them alone, and it’ll be okay.

Patrick Honner is an award-winning math teacher at Brooklyn Technical High School. He writes about math and teaching at and is active on Twitter as @MrHonner.

This article was published in the February 2014 issue of Math Horizons, along with more of Patrick Honner's interview with mathematician and author Steven Strogatz. Yet more of the interview is available online as a supplement.