Pam asks, "How can we as teachers encourage the flexibility and generalized use of strategies discussed above so that students can reason abstractly and quantitatively?"
Perhaps the best way to encourage flexibility and abstraction in mathematical thinking is to move beyond the tyranny of the Base 10 number system, which, after all, is simply an anthropocentric artifact of our having 10 fingers. I propose Base 2 for starters; then Hexadecimal. These, after all, are the number systems of our 20th and 21st century computers, and, therefore, of 21st century math skills.
Of course, once students have done these problems in Base 2 and Hexadecimal, they can relate their solutions to the anthropocentric counterparts in Base 10. As Eric points out, "it's just wise to take a problem and look at many different ways to approach it and then compare and consider those approaches."This is the comment I left two days ago on a blog post over at heinemann.com. (Hat tip to Barry Garelick). Given some of the some of the other comments in the thread, I was perfectly happy to look like a total lunatic to those who didn’t detect irony.
But then the more I thought about it, the more that irony started escaping me.
Maybe bases are the best way to encourage the flexibility and abstraction in mathematical thinking that so many math education experts say they want. What better way, in particular, to teach the concepts underlying the place value system? What better way to illustrate the abstract structure underlying 1, 10s, and 100s places than to compare this to the structure underlying the 1s, 2s, 4s, 8s, 16s, 32s and 64s places? Or to compare the decimal system—the tenths and hundredths places, etc.—to the halves, quarters, eighths, sixteenths, thirty-seconds, and sixty-fourths places? Not to mention myriad other possibilities, like long division in base 8, repeating decimals in base 3, and fraction-to-decimal conversions in base 7.
I loved learning bases back in 6th grade in France. I still remember the revelations they gave me about the “anthropocentric” arbitrariness of the Base 10 number system and of the gorgeous abstractness of number systems in general. And I’ve tried to share that passion with my kids—most successfully with J.
Why does no one teach bases anymore? Historically, they’re associated with the garbage of Sputnik-inspired New Math; mathematically, they require training that most teachers don’t get; pedagogically, they’re challenging to teach—and (for all the child-centered explorations they might inspire once taught) require the kind of direct instruction that’s been going out of fashion for decades
On that note, here’s some preliminary direct instruction in Base 8 from the Master:
Note the aside about today’s school kids (“the important thing is to understand what you're doing, rather than to get the right answer”). A reminder that the troubling trajectory of American math education is over a half a century old now.