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Viewing: Blog Posts Tagged with: Abstract Thinking, Most Recent at Top [Help]
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1. Antiquity Corner: Do You Have the Right Stuff for Time Travel?

If time travel were a reality and you could travel to the past, would you do so? More importantly, would you be permitted to do so? Think of the dangers to yourself and to the world. But let’s back up for moment.

It was Einstein who first theorized that time is a river and that one should be able to travel on it either forward or backwards. Even before Einstein, there was H.G. Well’s famous novel The Time Machine (1895), memorialized in fantasy films by Hollywood. Wells wrote at a time when an unwavering belief in technology as the means of building a better world was in vogue. Einstein, along with Nils Bohr, Enrico Fermi, Edward Teller, and J. Robert Oppenheimer led us into the nuclear age and the belief that science would make all things possible. No one, however, has advanced from theorizing about time travel to making it a reality. And perhaps that is a good thing.

People who have expressed to me a wish to travel back in time to visit one era, or one civilization or another, have done so without any thought of the difficulties. First comes the rigorous preparation in language, culture, and technology which would be required. Only scholars with considerable knowledge of the era under investigation could accomplish it. It would be too much for a layman. Also, proper clothing, documents, and currency, or precious metal that could be used as currency, would be needed, and possibly weapons and the training in their use. Some of these things have been made by historical reenactment groups. Even they, however, would be at risk in the cultures that they seek to emulate. The physical dangers could not be underestimated. The time traveler would have little protection against the casual violence of many ancient societies. Nor would reliance upon ancient justice systems be of any help. Assault, imprisonment, and enslavement would be real possibilities in places such as imperial Rome, the Byzantine

3 Comments on Antiquity Corner: Do You Have the Right Stuff for Time Travel?, last added: 1/26/2011
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2. Big Nothing: The History of Zero

While rocking out to Patti Smith, in celebration of her victory winning the National Book Award, I rediscovered her tribute, “Radio Baghdad.” The song celebrates the Iraqi city’s rich cultural and intellectual history, and as a refrain she specifically mentions its involvement in the invention of zero: “We created the zero/But we mean nothing to you.”

Smith honors Baghdad’s intellectual contribution to the establishment of zero as a number. Zero deserves her praise for its usefulness as a placeholder (as in the number 306), for its role as the additive identity element (if you add zero to any number, you get that number—in symbols, n + 0 = n for any number n), and for its contribution to the development of calculus. As the late writer David Foster Wallace elegantly claimed, “The invention of calculus was shocking because for a long time it had simply been presumed that you couldn't divide by zero.” Zero is a game-changer, a distinct value, and the barrier between positive and negative.

The richly informative book 100 Greatest Science Inventions of All Time tells the story of Al-Khwarizmi. In 810 A.D., this famous Baghdad mathematician convinced a group of fellow scholars that zero must be a number by demonstrating that zero behaves like a number when subject to common operations. Not only did Al-Khwarizmi thus effectively demonstrate zero as a number, but he also established himself as the founder of algebra. 
I love this story because I think it eloquently demonstrates the following dispositi

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3. Direct Instruction, Part 2

Teach strategies that are “generalizable,” i.e., that apply to a broad range of problems.
I’m not so sure about this one. Designing Effective Mathematics Instruction gives as an example that using a single pizza pie, cookie, or a pie chart (emphasis single) to teach fractions is a bad idea because any single model doesn’t apply to mixed fractions. Instead, the book recommends that you teach proper and improper fractions at the same time.

As I said above, I’m not so sure about this one. The underlying assumption is that students can master both types of fractions at the same time. But shouldn’t you teach pre-skills first? (Isn’t proper fractions a type of pre-concept necessary for understanding improper fractions?) I have the feeling that, in practice, if you try to teach both, you’ll end up focusing on proper fractions first.

I would agree wholeheartedly with this tip if we learned like computers. But we don’t. We humans like to attach meaning to what we do. We work best when we understand underlying concepts. So yes, teach generalizable, efficient strategies but keep Tip #1 in mind. Depending on your class, you may have to teach a less efficient, less generalizable strategy first so that your students have the opportunity to learn underlying concepts.
But what do you think? Leave your comments below.

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