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The Octopus Files: February, 2011

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Currently, the typical American high school student has the TI-8* calculator so tightly wedged in her hand that it borders on qualifying as a piece of biotechnology.

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Prologue: I spent the weeks surrounding Christmas in just a few hours from Icelanta, enduring bleak days without access to heat, internet, alcohol or automobile. Feeling like you live in a frozen bottle of white-out (modulo the fumes), you dig deep for psychological and physical reserves, working deep in Stewart's Calculus and the bottom of your DVD collection. It  turns out S. Darko isn't as bad as widely reported, and I have to wonder if some of the invective against it is rooted in some sort of  collective Donnie Darko-audience-anxiety about attractive young women illegally crossing the border into psychological science fiction. Take my opinion with a grain of rock-salt: by the time I watched it, my bones were so chilled they felt like some kind of rock-hard hallucinogenic ice cream. How does anyone live in the parts of Russia where winters like this are commonplace? No wonder they were so eager to get into space.

Speaking of Russians in space, President Obama's most recent State of The Union Address (January 2011, in case this gets garbled and sent to the wrong part of spacetime) struck a chord calling for greater emphasis on science and mathematics. It made me think. Politicians are well-known for endorsing things they can't practice themselves, and I have to wonder how many of the executives who call for greater mathematical literacy could actually demonstrate any. Forget the birth certificate, Mr. President, I want to see you take the derivative of a function (using the definition of a limit, not a shortcut). I have the feeling that the next presidential debate would come to a grinding if the moderator were to ask, "Is there a smallest positive real number? Explain."

I'm done making fun of politicians and on to serious questions. What sort of mathematics would we like to see taught? Currently, the typical American high school student has the TI-8* calculator so tightly wedged in her hand that it borders on qualifying as a piece of biotechnology. The focus is almost entirely on preparing for high-stakes multiple-choice tests (whether End Of Course, Advanced Placement, or SAT), which means they are presented in every question they encounter. These policies are ultimately okayed by elected officials or their designees, so it is legitimate to ask how prepared any of them are to actually address what preparing a student to engage in meaningful,21st century mathematical thinking would entail.

Believe me, if we developed an environment where that type of thinking were a survival skill (as it really is), students would develop the ability to do it. 

Children are forced to go to a place, memorize arcane rules and formulas with little explanation, with "real-world examples" so contrived and transparent that students should feel insulted they are expected to believe them. Is it any wonder that so many hate mathematics?

The first step is that mathematicians begin actively evangelizing, working to dispel the notion that mathematics is an exclusive activity inaccessible to the common person. The second is that anyone who professes to value mathematics (whether in the classroom, the Oval Office, or the Octopus Files) be prepared to demonstrate that feeling via meaningful action.

High school graduates in many states are required to pass three courses in mathematics. How many would be prepared to meet the following challenges?

• Can manipulating a ball of clay change its volume? What about its surface area? Explain.
• Using a map, approximate the perimeter of the coastline of California. Compare your answer to the actual answer.
• Develop and justify a method for multiplying binomials.
• Is there a largest positive real number? What about a smallest positive real number? Justify your answer.
• Several arthritis pain relievers utilize menthol as the active ingredient. Which one at your local pharmacy offers the most menthol per dollar?
• Is a piece of paper a two-dimensional or three-dimensional object? Measure its height.
• Find the average beats per minute of the current Billboard Top 40. Compare this to the chart of 10, 15, and 20 years ago. (Hint: You may count the beats yourself, or use another source.)
• Determine (without a calculator) whether or not a set of polynomials have a root (Hint: Use the Rational Roots Theorem, the mod p Irreducibility Test, the Intermediate Value Theorem, and theorems bounding the zeros).

Feel free to use outside resources, properly cited so as to avoid plagiarism. Please feel free to submit solutions or questions to me at andrew.octopus@gmail.com

If you find yourself struggling with these questions, the good and bad news is it is that you're not alone. If you're reading this, I feel certain that you're intelligent enough to answer these questions, but it's likely that you've never been trained to. Contact me and I will be happy to talk to you about this or other issues. What we need is a discussion, not an executive order. Now is the time to start.

-Andrew Octopus
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