Wednesday, March 4, 2015

Bases for conceptual understanding

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 (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.


James A said...

In the old days, (pre 15 February 1971) there were 12 pence to a shilling, and 20 shillings to the pound, 12 inches to a foot and 3 feet to a yard, and sixteen ounces to a pound, and ... Different bases were simply the air we breathed.

C T said...

Ah, you convinced yourself with your own eloquence when you wrote the ironical comment.
A person must be careful with what he/she writes. Long ago, I didn't want to go to law school, but I applied at my mother's urging; after I wrote an application essay about "why I wanted to go to law school," I had changed my mind and genuinely did want to go. ;)

Anonymous said...

I am all for teaching different bases for enrichment. However, there is actually a school of thought that Americans are not proficient in math facts because we persist in using the English system. Most people cannot calculate the ounces, and inches, etc in their head and they don't find mind math practical. Whereas in places with metric system, mental math are more useful because everything is base 10. Chinese speakers particularly had it easy because their 11 and 12 literally says ten one and ten two, make understanding place value easier for kids as well.

GoogleMaster said...

@Anonymous, I doubt that switching wholesale to the metric system would magically enable Americans to perform mental calculations. After all, our money is metric, but very few people can calculate a 15% tip in their heads, and many can't even do simple change-making (i.e. subtraction of decimal numbers).

Mnemosyne's Notebook said...

I remember learning bases back in sixth grade also - though in Bakersfield, not France. I think it's sad that there is nothing about teaching non-base 10 in the CCSS. Part of the dopey jargon is to prepare students for the 21st century - but the first time they'll see binary or hex is when they take an intro Computer Science course as a freshman? Granted, some of the better high schools will have CS classes, and I suspect those bases are taught then. But this is basic math. Leaving it up to a CS class is like punting fractions to Home Ec/Life Skills (Sally, the recipe is for 4 people, how would you modify it for only 3 people?).

I suspect that even the chowderheads who made the CCSS realize that most sixth grade teachers today would just die if they had to teach something as arithmetically intense as translating between bases.

Katharine Beals said...

As far as alternative bases in daily life go, we still have (besides the American non-metric system) seconds, minutes and hours; hours, days, and weeks...

What's challenging is that we represent these things using the Base 10 number system: it's the clash between the underlying number system and the number system used to represent it that makes the arithmetic especially messy.

Of course, it's also messy to use dozens of additional symbols, which is what you'd need whenever the underlying number system has place values involving powers of numbers greater than 10 (e.g., the powers of 60 seen in seconds, minutes, and hours).

lgm said...

Bases may be taught here as an aside when Roman Numerals are explained in 6th SS or scientific notation explained in 7th accel science. Really though, students who grow up in households with parents who involve them in upgrading computers dont seem to have an issue figuring binary out on their own....the rest will probably never get to the point of looking under the hood, so a lesson on bases would be wasted time given their lack of interest and issues with mastering fractions.

Auntie Ann said...

My niece at least know how to count in binary. I got her started by showing her the joke: there are 10 types of people in this world, those who understand binary, and those who don't.

From there, she learned the rest.

Anonymous said...

I learned bases in 6th grade as well--in Pasadena, CA in 1979. It was the most interesting thing I had learned in math...well...ever to that point.

Auntie Ann said...

the tyranny of Base 10? Oh, my.