# Does mathematics have a one percent problem?

I got whiplash from a series of math-related stories in the news yesterday.

First, I read the inspirationally-titled "National Competition Shows that Math Smarts are Alive and Well in America." The article describes the most recent installment of Moody's Mega Math (M3) Challenge. The M3 challenge is a high school applied math competition sponsored (obviously) by Moody's, the credit rating and financial services corporation, and by the Society for Industrial and Applied Mathematics. In the competition, five-person student teams have one day "to solve an open-ended, realistic, applied math-modeling problem focused on a real-world issue." This year's challenge was to determine the best U.S. locations for establishing high-speed rail lines.

My spirits soared over the strong mathematical future soon to be built by our nation's youth. Or at least, they soared for about two seconds, until the universe smacked me back down to earth with the second article, "Parents' poor math skills may lead to medication errors." Here's the horrifying scoop. It's long been known that poor reading skills can lead to dosing errors. But a study recently presented at the Pediatric Academic Societies conference examined whether poor math skills play a role too. Investigators engaged 289 parents who had brought their children (under eight years old) to a certain pediatric emergency department. The parents were given a reading test and a math test, and were asked to dispense some medication. The results showed that

Nearly one-third of the parents had low reading skills and 83 percent had poor math skills. Twenty-seven percent had math skills at the third-grade level or below. 41 percent of parents made a dosing error. Parents' math scores, in particular, were associated with measuring mistakes, with parents who scored below the third grade level on the math test having almost a five times increased odds of making a dosing error.

One of the investigators commented

Dosing liquid medications correctly can be especially confusing, as parents may need to understand numerical concepts such as how to convert between different units of measurement, like milliliters, teaspoons and tablespoons. Parents also must accurately use dosing cups, droppers and syringes, many of which vary in their measurement markings and the volume they hold.

I am no statistician, but from the brief description of the sampling scheme, I could certainly conjecture that any number of confounding variables could be at work here. Still, after reading this article, I was left wondering how our country has any live school-age kids to speak of given the massive ibuprofen and antibiotic overdoses that we are assuredly inflicting on them.

Reading these two math-related articles in quick succession made me wonder: do we have a one-percent problem when it comes to mathematics? We certainly have a mathematical elite, and we certainly have many who struggle. Do we have a healthy mathematical middle class?

One possible solution for raising math abilities in this country, as inadvertently suggested by yet a third math-related article I read yesterday, is simply for us to all get bashed in the head and become geniuses.

# Alpha dog

Let's talk about the unavoidable Stephen Wolfram.

The mathematicians and non-mathematicians among you will likely have heard of Wolfram. He's a physicist, mathematician, and computational scientist who developed a useful bit of software called Mathematica for numerical and symbolic computations. He's also a MacArthur Genius and the presenter of one of the most popular TED talks. Much earlier in his life, when he was 20, he became the youngest-ever recipient of a Caltech Ph.D., a status he has retained by a hair over my former undergraduate student Catherine Beni.

In 2002, Wolfram published his book A New Kind of Science, which sought to explore the nature of computation and its relationship and applicability to the real world. The scientific community received this book with mixed feelings. Most mathematicians I know are quite critical of the book, both for its apparent egotism and lack of rigor. I think it's fair to say that the academic math community considers the book to be a swing and a miss.

In contrast, Wolfram's real home run is his computational knowledge engine Wolfram Alpha, launched in 2009. I can't urge you enough: go play with this amazing tool. You can ask it a huge range of questions related to mathematics and, more broadly, data of many different sorts. While most mathematical and computational tools require input to be given in a particular syntax, Alpha uses natural language processing to try to figure out what you are asking it. You don't need to learn a programming language or any particular querying commands.

To get a sense of what Alpha can do, I challenge you to try to find:

• The derivative of $x^y ln(xy^2) + 3e^{cos(x/y)}$ with respect to $y$
• When the next solar eclipse is
• How the U.S. climate has changed from 1900 - 2011
• What a fractal known as the Pythagoras Tree looks like over 10 iterations
• How many days you have been alive
• What the geographical distribution of life expectancy in Africa looks like
• What year Henry David Thoreau's Walden was published
• For how many years Flintstones and the Jetsons overlapped on the air
• The Scrabble score for the word exogenous
• How many medals Michael Phelps has won
• How Netflix stock is doing lately
• How much a \$250,000 mortgage at 5.5% over 30 years costs
• What demographics in Egypt are like
• How long it would take a 44 year old male weighing 90 kg to lose 17 kg if he eats 1500 calories per day
• Where the first flight out of Chicago went to today

In the interest of keeping this post accessible, I've focused on general knowledge in the list above, but make no mistake about it: Alpha can do serious math.

Much like they decried the advent of calculators, some mathematicians and mathematics instructors decry Alpha for how it will allegedly ruin mathematics education. With a freely available, Internet-based tool that can differentiate, integrate, solve differential equations, compute eigenvalues, make graphs, and do so much more, what is the motivation for students to learn symbolic mathematics?

While I don't deny that symbolic skill is necessary for people who will become mathematicians, I nonetheless vehemently disagree with the old-school outlook above. Wolfram Alpha is going to ruin math education no more than the advent of calculators did, and no more than abacuses did. Computational tools are a natural, healthy, and useful part of civilization's mathematical progress. Consider this: the basic algebraic knowledge an educated person has today constituted advanced, cutting-edge research in ancient times. No one cries about the fact that we no longer do mathematics the same way as the ancient Egyptians, Greeks, Babylonians, and Mayans. Computational tools free us to think about ever-more advanced and complex ideas, and ideas are what math is really about.

# Know thyself

How well do you know yourself? How accurate is your perception of what you know? To what extent do you understand how you learn? Whether you are a student or instructor, these are among the most pressing questions that you need to consider. I want to give you a handle on how to do so.

Defining the essence of  knowledge is a tangled, murky, and worthwhile endeavor that has busied philosophers for many centuries. In the context of modern education, though, it is as much a tactical question as a philosophical one. Teachers and learners need to disaggregate different types of knowledge. The famous (infamous?) Bloom's Taxonomy makes one suggestion for how to do so through its identification of different domains of learning.

I am a bigger fan of Anderson & Krathwohl's taxonomy, which is a revision of Bloom's. In a later post, I will discuss this taxonomy and how to use it, but for now, suffice it to say that an important process that it highlights is metacognition, essentially thinking about thinking. For those of you who are fans of such self-referential ideas, you may also appreciate this:

http://xkcd.com/688/

Metacognition is a crucial aspect of learning, and in my experience, one that we frequently neglect. Metacognition has several aspects, but two important ones are

• Metacognitive knowledge, which is what a learner knows about her/himself, and
• Metacognitive regulation, which refer's to a learner's ability to react to metacognitive knowledge by making alternations to control her/his own learning.

What does weak metacognition look like? Suppose that I assign calculus students to read a textbook chapter on derivatives. I ask my students to write a brief response to the reading in which they explain what they found challenging. A response like "I don't understand derivatives" or "derivatives are confusing" signals a student with weak metacognitive knowledge who cannot identify particular points of difficulty. A response like "I don't see why one of the correct geometric interpretations of the derivative is as the slope of a tangent line" signals a student with stronger metacognitive knowledge who can articulate specific challenges. Hopefully, this metacognitive knowledge serves as the gateway to metacognitive regulation, which would address the subject matter. For instance, the student could then decide to re-read the specific paragraph explaining the concept, try a practice problem involving the geometric interpretation of derivatives, ask somebody for help with the idea, or take one of dozens of other possible concrete steps.

If you are a student, I encourage you to work hard to be clear and specific with yourself about what you find challenging. If you are an instructor, I encourage you to discuss metacognition with your students and to structure activities and assignments that require it. For the sake of concreteness, here are two examples of how I do this:

1. As I mentioned above, I ask my students to respond to reading assignments (in writing) and to articulate their difficulties. I give them feedback on whether they have expressed their difficulties with sufficient specificity.
2. I give students the opportunity to partially regain missed points on quizzes by turning in corrected solutions. But as part of these corrected solutions, they must also include an explicit discussion of what they did wrong the first time through, and what their original challenges/misunderstandings with the question were. I only restore points if the new solution is correct, and if the metacognitive explanation is well-developed.

Good metacognition take practice. We can all learn to be better metacognitive thinkers. I wish you luck in your quest to know yourself.

# Pedagogy vs. curriculum eXtreme smackdown

Man vs. nature. Red Sox vs. Yankees. Mothra vs. Godzilla. Our world is rife with conflict. The one I want to focus on today is pedagogy vs. curriculum.

Pedagogy refers to the art and science of teaching. Curriculum refers to the material taught. Put simply, curriculum is the "what" and pedagogy is the "how." While the line between pedagogy and curriculum can be blurry at times, it's a crucial distinction nonetheless.

Within the world of college/university mathematics, curriculum has pummeled pedagogy into a bloody pulp. I've observed mathematicians engaging in discourse on teaching all over the country, and 99 times out of 100, these conversations are about curriculum. With good intentions, we obsess over "covering enough material." We worry about the sequencing of our departmental classes. We strive to find new and interesting mathematical examples, homework problems, and projects to incorporate into our classes. These are laudable efforts, but they only get at half the picture.

If you are an instructor, I want to challenge you with a question: Just because you cover something, does that mean that your students now know it? Educational research has for decades told us that the answer is no. (Here's a classic example from "A Private Universe," showing graduating Harvard students unable to explain what causes the changing of the seasons.) And yet this notion seems to underlie much of what still goes on in today's college mathematics classrooms.

The neglect of pedagogy relative to curriculum at the college level is hardly surprising. The deans, department chairs, and instructors are trained experts in mathematical disciplines, and not in teaching. But just like Sherlock Holmes needed his foil of Moriarty to thrive, curriculum needs pedagogy to have its full potential unleashed.