Neither Borrower Nor Lender Be…

Despite its name, the Sapir-Whorf hypothesis is not something found on Star Trek.  Instead, it’s based on the following idea:  How you speak influences how you think.  I stress this concept in most of the courses I teach, but nowhere is it more important than in elementary mathematics.

Consider the following subtraction:  42 – 27.  If you were taught the standard algorithm, you might have been taught that since you can’t subtract 7 from 2, you have to borrow 1 from the 4, making it 3; the 1 then makes the 2 into a 12, which you can subtract from.  For generations, schoolchildren have been taught to subtract by borrowing.

And for generations, schoolchildren have been taught something that is blatantly false!

If I borrow your car, you expect I’m going to return it to you.  You’d be very unhappy if I kept it, or sold it for scrap.  Yet this is exactly what happens to the 1 “borrowed” from the 4:  it is never returned.

A better term is trade or unpack.  Let’s shift gears a moment, and suppose you need 8 eggs.  You go to the refrigerator, and find to your dismay that you only have 4 eggs, and 1 dozen.  You can’t get 8 eggs, because you only have 4 eggs.  Fortunately, you’re clever and realize “Hey, I can open up the dozen, and I’ll have 12 eggs.”  Indeed:  from the dozen, you’ll take 8 eggs, leaving 4 behind.  (The refrigerator will still have the other 4 eggs, which weren’t touched, so all together you’ll leave behind 4 + 4 = 8 eggs, possibly in two cartons…a frustratingly common situation in our house)  You can watch this video to see this exact process.

What about 42 – 27?  Here you have 4 tens 2, and want to subtract 2 tens 7.  So you unpack one of the tens, giving you 3 tens 12, from which you can remove 2 tens 7, leaving 1 ten 5:  15.

What about 4 hours, minus 2 hours 27 minutes?  Again we unpack:  4 hours is the same as 3 hours 60 minutes, from which we subtract 2 hours 27 minutes, leaving 1 hour 33 minutes.

What about 2 miles minus 153 feet 8 inches?  I dunno…that’s the “English” system of units.  You can tell how useful the system is, since (by GDP) two of the three most powerful countries to use it are Myanmar and Liberia, which most people would be hard-pressed to find on a map.

Fortunately, I collect useless trivia, so I know 1 mile = 5280 feet, and 1 foot = 12 inches.  So unpack 1 of the 2 miles into 5280 feet; then unpack one of the feet into 12 inches.  This means 2 miles becomes 1 mile 5279 feet 12 inches, from which we can remove 153 feet 8 inches, leaving 1 mile 5126 feet 8 inches.



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