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## Wednesday, May 13, 2015

### Time to Reverse Direction

I have read a lot about people blasting Common Core for de-emphasizing, if not leaving out, the old right-to-left "borrowing" (or "renaming") method of subtraction.

To all those people I should first point out that even on a calculator, numbers are entered left-to-right. Having students learn right-to-left for paper calculations, but left-to-right for electronic calculations, has helped create a generation that finds mental arithmetic almost totally impossible.  Conflicting methods all through school create havoc in the student's brain.

Imagine learning to write right-to-left. Quickly, spell your full name backwards.

The Common Core recognizes that teaching right-to-left subtraction does not promote understanding of what subtraction really is.

All over the web there are people blasting supposed new and different techniques for subtraction. Business Insider had one last October (see it here). In it the author, Andy Kiersz, writes
The "counting up" method (which is what's depicted in the textbook above) is not intended to replace the standard way. Instead, it captures some of the underlying aspects of subtraction and place value that allow borrowing and carrying to work.
His comment bothers me. It is based on the assumption that the old methods are fine, just in need of a facelift.

In fact, they were not fine. The whole mess of right-to-left arithmetic needs to be tossed. We say numbers left-to-right, we write numbers left-to-right, we enter them on calculators left-to-right, we even say, remember, and dial our phone numbers left-to-right. Why would anybody think that we should be calculating with them right-to-left?

Another article referring to the same example is by Ed Morrissey, also from last fall (see it here). In it he says "the “counting-up method” requires a paper calculation and more complicated cognitive judgments than simply subtracting and carrying over."  That analysis is highly debatable. Anyone grounded in left-to-right addition finds the 2+60+200+25 fairly easy.  The process does not require paper calculation at all. In fact the need to write down paper calculations will diminish as one gets more familiar with left-to-right arithmetic.

Thankfully, students are beginning to learn arithmetic in a manner that will help them internalize the concept making them stronger and better thinkers. The "New Math" back in the 60's tried to do that, but it never dealt with the "right-to-left" vs "left-to-right" issue. Back then it was apparently presumed that if people obtained a better number sense the old arithmetic methods would make sense. That was still climbing the uphill battle because right-to-left was against-the-grain.

The examples referred to in Kiersz' article begin to make full sense to someone who has begun to think and work with numbers left-to-right. In the "subtract 38 from 325" question, I would subtract 40, get 285, then add 2 back to get 287.  It is different from "counting up", but it's much easier to understand than "you gotta take 8 from 5 but you can't take 8 from 5 so you take the 2 in the tens place, make it 1, and and a 10 to the 5 in the ones place making 15,.....". (Apologies to Tom Lehrer).