Life of a Rooster

Memoirs of a psychiatrist, journalist and educator

Calcul and Arithmetique

on March 15, 2014

As promised, I shall give credit where credit is due. The French education system in the elementary grades is excellent in some aspects. I shall address here the subject called “Calcul” (Computation) in the lower elementary grade, and Arithmetique in the upper elementary grades.

french subtraction with regrouping

I cannot say that they were good at making us learn to calculate, because the Chinese are vastly superior in that aspect (more on that later). But there are a few points which are taught very differently than on the American and Asian continents. The first one is the way they regroup during subtraction, which is sometimes called “borrowing” a ten.

In Taiwan or the US, somewhere between Grades 1 or 2, all children meet for the first time subtraction with two digits, and all of a sudden that number in the second row has a unit digit which is smaller than the one above, and the child wonders how to deal with this problem. For instance, you may have 42 – 18 = ? How do you take 8 away from 2? Good teachers of course, teach how to do this, before they confront their students with this question. But little children have a tendency to disconnect things from each other, and may not remember that those blocks they were playing with the day before had some relationship to the problem they are meeting today. Some children, to this day, are very creative and expedient. Well, just turn things around. Take the 2 from the 8 and you get a 6. Oh, why is it wrong?

But I diverge.

How to regroup or "borrow" a ten in subtraction, according to the American method.

How to regroup or “borrow” a ten in subtraction, according to the American method.

So, the regular non-creative American child will cross out the 4, write a 3 above it, add a small 1 next to the 2, which makes it a 12. Now you can take 8 away from 12, which yields a 4. Then 3 – 1 = 2. Result: 24. Very good. How do the French do this? The maitresse back in Grade 2 told us to borrow a ten from the neighbor above (the 4) and write it as a small 1 next to the 2. So far, it is the same as in the American method. But do not cross out the 4 and do not write that 3 above the 4. Then, “what do you do when you borrow something? Yes, you must return it. So now, let’s return the ten to the neighbor below.  Let’s write a small 1 next to the 1 of 18. 12 – 8 is 4, and 4 – 2 = 2. Answer: 24.  Any questions?”  This was before I had turned pathologically shy. Maybe this was one of the factors that did turn me pathologically shy. Whatever the case, I raised my finger (in France, you raise an index finger, not the whole hand). “Yes?” asked the maitresse.

How to regroup or "borrow a ten" in subtraction according to the French method.

How to regroup or “borrow a ten” in subtraction according to the French method.

“If we borrow from the neighbor upstairs,” I asked, “why are we returning the ten to the neighbor downstairs? Shouldn’t we return it to the neighbor upstairs?” Does it not make sense to you, my reader? When you borrow sugar from your neighbor Mrs. Smith, you do not return the sugar to Mrs. Jones. You might make Mrs. Jones very happy, for she just got some free sugar, but that is not the point. I mean, I really truly wanted to understand this thing.

The maitresse stood still for a moment and glared at me. She did not look happy at all. She took a step forward and stared hard. Slowly, with buried roars in her voice, she asked, “What-do-you-do-when-you-bor-row-some-thing?”

“Er… we.. we… er… return it?”  I was cowed into stuttering.

“Precisely. A-ny ques-tion?” she asked with threatening thunder.

“No. No questions.” That shut me up for good. Did I dare have any more questions?

Today, as an educator and a trainer of educators, I can see that she did not know why. She knew HOW to do it, but not WHY. And when a student shows you up, you take revenge by terrorizing them. I can tell you why, dear readers, if you wish to know. The answer to a subtraction question is called a “difference” for a very good reason. It indicates the difference between two quantities. So the difference between 3 and 1 is the same as the difference between 4 and 2. Instead of making the upper digit smaller, you make the lower digit larger. Same difference.

So why should the French method be better? For something simple like 42 – 18 = 24, it might look as if both methods require an equal amount of time, intelligence and effort. But let’s look at something like 40,102 – 15,849. American method: You want to take 9 from 2, but cannot. You want to borrow from the neighbor on the left, but oops, it’s a zero. OK, no panic, go to the next neighbor on the left. Cross the 1, write a 0. Write a little one next to the 2 to make it a 12. Now, you can take that dratted 9 away from the 12 and get a 3. What next? Oh, OK, take the 4 from the 9 and get a five. Easy. Then, darn it, it’s another zero. Go to the neighbor on the left, crumbs! It’s another zero! No panic, no panic. Go further left, ah… here’s a good digit, a 4! Cross out, write a 3, put a small 1 next to the zero to make it a 10. Now what? Oh, OK, now cross out the 10 and write a 9 above it. Where was I? Oh, yes. Go back to the 100s column. 8 taken from 10 is a 2. Then five from 9 is a 4 and finally 1 from 3 is a 2. Is your head spinning by now? How many of you tried doing it on your own before reading all this? How many of you made no mistake at all? Really? Try 201,010 – 98,999. OK, be truthful now. How many of you made no mistake at all and had no hesitation at all? Now imagine a small child, especially one who is a bit hyper and has a very short attention span and has a messy handwriting. Do we really expect this child to do this complicated manoeuver with no mistake whatsoever? Is our goal to see how to trip him or is it to help him?

Compare the clean lines and clear view of your numbers in the French method. The child's mind is equally clear while performing this operation.

Compare the clean lines and clear view of your numbers in the French method. The child’s mind is equally clear while performing this operation.

Let’s look at this same operation with the French method. Instead of traveling up and down the columns and across the tops back and forth, you only go down the columns one at a time. This is what you say to yourself: “Zero minus 9, cannot be, borrow return (write a small 1 next to the zero and another small 1 next to the 9 — neighbor below), 10 minus 9 is 1.” You now move to the second column and say, “1 minus 10, cannot be, borrow return (small 1, small 1), 11 minus 10 equals 1.” You move to the third column: “Zero minus 10, cannot be, borrow return (small 1, small 1), 10 minus 10 equals 0.” and so on. You only deal with one column at a time, you do not need to cross out or rewrite and build a pyramid of digits on top of one another. You need not fear zeros and ones, just work on one column at a time. That’s a successful method, one that can be used with confidence by any child, steady or hyperactive, regardless of what numbers you throw at him.


Ah...! The beauty of a streamlined subtraction!

Ah…! The beauty of a streamlined subtraction!

And so, while my own teacher scarred me when teaching it to me, I teach it to all my students so we won’t have a subtraction error sneaking into otherwise beautifully worked word problems or algebra problems. And I always ask them, “Do you want to know why it works? I can explain it to you if you want to.” And so far, none has been interested in knowing why.

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