Is "borrowing" the most appropriate term when used to subtract one number from another?

When I was younger, the term “carry” was used. For me, borrow sounds a bit odd.

Borrowing is how I was taught.

But I could see “sharing” or “carrying”.

“Stealing” would be a bit much.

Sure, use “carry back”. “Convert” would be best but is actually above a second grade vocabulary level.

But if semantics is messing you up, I would posit arithmetic is too complicated for you.

I have never heard a kid say, “when do we give the ten back?”

What about “abduct”?

I was taught “borrow” and never gave it a thought. “Carry” was used when we were adding.
45 – 17 = Make the 5 a 15 by borrowing a one from 4 to make 3 15 – 1 7 = 2 8.

When adding 45 + 17 = We make 5+ 7 = 12 leaving 2 and Carry the 1 to the tens column to make 1 + 4 + 1 = 6 2.

As @LuckyGuy was instructed, so was I: we “borrow” from the column to the left of the current operation for subtraction, and “carry” the digits past the single-digit sum to the left when adding.

But I like the idea (especially while teaching an adult) of “stealing” from the columns to the left – it would make that a more memorable operation. In keeping with that theme, then, how should we “carry” to the left while adding? “Hijack” – my preferred verb – is too close to the “stealing” we’ve already discussed in subtraction. Maybe “throw off” to the left?

No. Borrowing is when you take something and lend it to another person or place. Subtracting is simply taking away without regard for where it ends up.

I thought about it some more, and I can sort of see a way to justify talk about borrowing, but I still think it is misleading. For 34 – 6, we imagine that 34 is a member of the mathematical universe. Then we can say that for the purpose of this problem we are allowing the 3 to shed one of its units, but the number 34 will return to its pristine form.

I was taught that it’s called “carrying” also.

Although, I am all for “stealing”. Although I’d actually rather it be called “Embezzling”

“Now you must embezzle the 2 from the 5”

Borrowing isn’t a very good term. I seem to recall even when I was first taught it, that it wasn’t the right word to use. I think the teacher may have even said something about it being just a peculiar metaphor.

It sort of works in the sense that it convey that we’re taking something from what the other number represents, and that it’s only a temporary step that we’re about to clean up after we’re done with the step we’re on. The peculiarity of the word can be useful because it’s an odd label that refers to one peculiar thing.

I imagine there could be better words used, and wonder why/if one hasn’t already been found. Math teaching vocabulary has changed a bit over the years, though it seems like there’s a bit of an unfortunate emphasis on artificial methods rather than concepts, at least with the kids I’ve tried to help with their math assignments (which I’ve seen getting in the way of understanding the concepts underneath).

For subtraction, I think of it as taking 1 from the digit to the left, which always means taking 10 in the units of the digit you’re considering, so even writing a little +10 overhead would make sense. I’d call it taking, or moving, and/or converting to 10.

Well, teaching an adult is different than teaching students. I’d say “Now, we need to borrow one from the neighbor.”

Borrowing is fine with me. These are numbers we’re talking about and not some living thing.

If you think that’s bad,you should see the “Common Core” math. They have to learn a whole new vocabulary, such as “regrouping” the numbers. In 32 – 6, they first learn that 20 + 12 is the same thing as 32, so now you can subtract 6 from 12 and add back the 20. It’s not very complicated but increases the steps to the point of confusion.

It’s just a metaphorical adaptation of a literal expression to a specialized situation. In that context, “borrowing” does mean literally what you are doing when you take ten from the next column in order to subtract smaller from larger. “Borrowing” is well established in the context of arithmetic and doesn’t need to be changed.

Thousands of words have more than one meaning or work in more than one sense. Just as we can still use “dial” for calling phone numbers on a keypad, with no actual dial involved, we extend the meanings of words by new applications, and then that is what they mean.

@Jeruba Isn’t it borrowing without giving it back, though? Why not just “taking” or “moving” then?

@Jeruba , Your point is well taken. I am not asking to have all the textbooks rewritten, but I was looking for a more useful metaphor. It was my first class with the student. She understood carrying for addition, but was having difficulty with subtraction. My strategy for dealing with that will be to take an addition problem like 28 + 6 = 34 and then show that subtraction is the same process in reverse. For addition we add 8+6, carry 1 and add it to the 2. To do 34 – 6, take the 1 from 3 that we previously added and combine it with the 4 to make 14. I kind of like talking about carrying back in place of borrowing. It is like carrying but moves in the reverse direction.

@Zaku, if you borrow my copy of Zen Mind, Beginner’s Mind and don’t return it, I’d say I gave it to you. But if you subtract 89 from 234 and have to borrow from the tens column to complete the computation, that doesn’t reduce the overall value of the result. Nothing is given back because nothing is really taken away. You’re just exchanging one “ten” for ten “ones.” I’d say it’s more like converting than borrowing, and certainly not like taking, just as when you ask for some small bills at the restaurant so you can leave a tip. The total amount of your change doesn’t decrease.

I think “borrowing” is a clear enough label for the concept that kids can get it right away; and once you’re an experienced subtracter, you can call it anything you like (as long as you’re aware that it’s nonstandard) and it’ll still work.

@LostInParadise, your strategy sounds good. Maybe a group of concrete objects such as colored sticks or pennies would make the reversibility concept easier to explain. Or how about an abacus?

If you use pennies for the ones digit (or one cent), you could also use dimes and dollar bills for the tens and the hundreds places (or ten cents, a hundred cents)—if I’m not making it too complicated or adding too many moving parts…? Then when you “borrow” or “steal” or “carry back” a dime, you could convert it into ten pennies. (And vice-versa with the addition, ten pennies become one dime.)

I learned it as “borrow one, pay it back” but the way I learned it is not the way my kids and grandkids were taught.

If I had the problem 327 – 59 = I would start out “borrowing one and making the 7 a 17 then subtract nine. The next step would then “pay it back” by making the 5 a 6 so the next step needed was to “borrow” one and make the 2 a 12, subtracting the 6 from 12. The final step was to “pay it back” by putting a 1 below the 3 and subtracting it leaving the answer 268.

My kids learned that you start out the same, making the 7 a 17 but then that changed the 2 to a one and subtracted 5. but this necessitate taking 1 from the 3 to make the hundreds digit a 2 and the tens digit 11 allowing 5 to be subtracted from 11. This left a two in the hundreds place with nothing to subtract from it. Still yields the same answer but makes the term “borrow” incorrect since nothing is going back.

Does this make sense?

@rojo That method actually does make the “borrowing” make sense, if you “pay it back” in that way, although it’s still a metaphor that it seems to me is about the procedure used rather than what’s actually happening mathematically. I like it because it follows through on the “borrowing” aspect of the metaphor, unlike borrowing it and not paying it back. However, it bugs me a little in comparison to the other, because you actually didn’t reduce the digit on the left when you borrowed.

Personally I’d prefer a way of working the problem that more directly models what is actually going on the math. I’d make sure all the students understand how a number represents a total which is broken into digits where each digit to the left means ten times the digit to the right, so the first number on the right is how many 1’s, the next digit is how many 10’s, etc. Then when it comes to “borrowing”, it’s not borrowing but taking one from the left digit and moving what it represents to the right digit, which means you reduce the number on the left by 1,and add it as 10 in terms of the number on the right. That concept can be explained to 8-year-olds if you take the time, especially if you have some physical models. If you can explain what’s actually going on, then the shortcut systems (to theoretically be able to solve problems sooner) are actually noise in the way of understanding the actual math.

How about covering?
I went to breakfast with a friend, and they were short when the bill came, so I covered him.
Same thing.

@rojo , I wonder if that is where the idea of borrowing originated. I agree with @Zaku that the process misleading, because the 1 that you are “paying back” actually means that you are taking away 1 more from the tens column.

@filmfann, That covering idea is interesting. It describes the action as proceeding from the tens column to the ones column, which is easier to understand.

I agree, the way my kids learned it makes more sense mathematically but, and perhaps it is because it was the way I was taught, the other way seems easier to actually complete the problem.