Standard Algorithm v. Austrian Subtraction
If you have to do the following math problem:
1811
- 969
assuming (just play along!) you can't do it in your head and don't have a calculator with you, how would you do it? Would you use the standard American algorithm, or "borrowing" as they explained it to us in school? Would you do it using Austrian subtraction? Steve Wilson, the math professor whose material on alternative subtraction algorithms I'm linking to, writes that the standard American algorithm is "pretty messy with all of the cross outs and rewrites. Even if you do all the cross outs and rewrites in your head so that your paper doesn't get messed up, its a lot to keep in your head."
Yes, yes, yes! When I was in elementary school, I found the mess generated by the "borrowing" method terribly difficult to handle, and I literally experienced anxiety attacks trying to deal with it. When I learned Austrian subtraction in sixth grade, it was an enormous relief, and it made so much more sense to me -- something about my cognitive style, I reckon -- and I've used it ever since. I don't know why I've been thinking about this; maybe it was spurred by a conversation I had with Jonathan last night about intelligence testing (and childhood experience with). At any rate, I'm curious to find out if any of you are Austrian subtraction advocates.*
* UPDATE: I forgot to point out that I'm really talking to readers in the U.S. here. In his description of Austrian subtraction, Wilson claims that it's the preferred and most-often-taught method in European schools (is he right?). Of course, any thoughts on either method of subtraction, or additional ones, are welcome.
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Hmm.
If I'm not wrong we were taught the 'normal style' of 'borrowing', something as shown in the standard american algorithm.
mathematical shortcuts
This is easy, really really easy. You don't need either algorithm and its faster amd you can do it in your head, anyone can. Of Course, these days, no one can make change except for maybe waitstaff. Can the impending collapse of civilization be far behind? :)
Simply round up or down to the nearest number, perform the operation of addition or subtraction, then recover the distance from the the original number and add or subtract appropriatly. Done!
See? easy!
In this case I instantly subtract 1000 from 1811 and to that result I add back in 31. Its quick, easy and I had the result before I was done reading the paragraph.
There are simular tricks you can use to ease your way through the multiplication of numbers larger than the multiplication tables we were taught in school. Basicly, you just multiply in chunks you can deal with, tens or hundreds or whatever and add the up those results to get your final answer.
Rounding up and down
Yeah, usually these days I do that* (I too have had jobs that required me to make change, and that's how I learned the method). But, if you play along and assume you couldn't do it in your head, which subtraction method would you use? :-)
* Edited to add: Actually, I do what's going to be fastest given the available resources. If I have a pen and paper, I'll do it the Austrian way, but if I don't, I subtract using the method Brenda describes.
Ja Wohl!
I had no idea these had different names, but I definitely think that the "Austrian" version is an easier approach. I had lousy handwriting as a youth, and I could never keep borrowing straight. When a teacher showed me the alternative method, I never went back.
When I was shopping for classes a few semesters ago, I sat in on a few of sessions of a Clay Spinuzzi class. For a demonstration, I volunteered to do a subtraction problem he put on the board. I think his point was something about standard approaches to problem-solving, which I munged by using the Austrian method, which the other class members thought was weird.
American English
Here in England, I seem to remember being taught the 'American' method, but I actually used the same method as brenda for the example - subtract 1000, and then add 31.
I don't remember whether we actually got taught that one, or I just picked it up from somewhere...
that's brilliant
much neater & cleaner. & i've never seen this before in my LIFE, which doesn't surprise me in that i had all my schooling in the states & never learned any subtraction in europe, but does rather surprise me in the context of my undergraduate education & licensure in middle school education--all subjects including math for 4th-through-8th grade. you'd think, in teaching teachers how to teach kids with lots of different learning styles & cognitive responses to things, they'd show us there's more than one way to do subtraction.
the only potential drawback i see is that for some kids, for whom the notation is very important & literal, it's going to be hard for each of those one's to do double-duty--especially when each one is both a one to add IN to another number and a one to add ON as a prefix (or are they explained as ones and tens?
Different algorithms
I went to primary school in Ireland and we were taught the Austrian version; though it's news to me that it's actually called "Austrian subtraction"... I wasn't aware there was another way of doing it. And having just looked up the American algorithm, it certainly seems less intuitive. Though that - of course - could be just down to a lack of familiarity.
Algorithmic Sleight Of Hand
If I were pressed, like if the number were too large, I would use the standard method. Only if i couldn't find a calculator.
The method I describe I learned by being behind a cash register but if you think about it, its the same method, the operations are simply ordered differently. What really helps me out is that I visualize a number line and my minds eye slides along it. Sort of like a mental slide rule.
I believe that this is how mentalists are able to perform operations on large numbers quickly and in their heads. My mind just has a simple number line on it. No logs or square roots and really big numbers slooow iiiit doowwwwn. I think I need to upgrade! :)
suffice to say....
....you should really add a new category for "mathematics." That would be very, uh, old school. lol.
I finally bit the bullet and switched to wordpress, C. The redesign over at marcoe is coming along. Unveiling to come. You should see the tanlines those MT shackles left on my ankles. Damn that rebuilding!
There are lots of different "
There are lots of different "standard" algorithms. The French & English systems here (or Protestant and Catholic, which did the same things till they were changed to language ones) use very different division methods, and woe is you if you change school systems. (I don't recall what the other method looks like -- sort of a mirror image of the one I learned.)
But when subtracting, I also usually use Brenda's method.
Tyra, you said: the only pot
Tyra, you said:
The way it was shown to us isn't exactly like the way it looks in Wilson's notes. Wilson gives us this image:
But when my 6th grade math group did it, we were instructed to put little subscript 1's under the bottom numbers too, like in the ones column of the example Wilson gives us, I would have the 1 beside the 2 like he has it, but I'd also put a 1 beside the 6 in 4568. So it ends up looking like:
That helped me. I've been remembering more about elementary school math, and I didn't mean to suggest that we were only just starting subtraction in sixth grade. Heavens, no. I do wonder why we didn't encounter the Austrian method until we'd already been doing borrowing (or as it's now called, regrouping) for years. Also, I was in the advanced math group, and we were the only group who learned the Austrian method (it was in our math group's textbook). You'd think the method might have been helpful to the other math groups too.
One more thing: Of the people in my math group, I was the only one who liked Austrian subtraction. Everyone else hated it, just learned it because it was going to be on the test, and then went back to the regrouping style.
Odd
That's funny, I was taught to use superscripts. But then, that was...ah hemmm... back in the early seventies. Probably a glacial advance in pedogogy.
New Math
In the early 1960's we were taught "New Math." You had to
write everything out and show the borrowing. "Doing it in
your head' was not only discouraged, but you got downgraded
for it.
As I was struggling to figure out how to subtract, my mother
tried to teach me the way she learned it, but the teachers
sent around little booklets, warning the parents NOT to help
their kids, because they weren't teaching them the right way
and everyone was getting confused.
I finally taught myself to subtract when my grandmother got
me a wristwatch. I used the numbers on the face to figure
out my own "method." But I still got downgraded because
I wasn't "showing" how I got the result.
I still can't do math. I use my calculator for everything and
I'm not kidding.
Anyone want to ask about all of the kids who were taught ITA?
That's what my sister, who came into the system a few
years later, was taught when they started reading. It was
a phonetics alphabet system with separate books, etc.
She was already reading the "regular" way and, again,
the teachers told the parents to stop butting in and
trying to confuse the kids.
My sister is now a lawyer and she can hardly spell her
own name.
I love education!
munging
And i still have not forgiven you. Yes, I was trying to discuss subtraction as embodied and mediated, a point that is really quite nicely made in this thread. BTW, I'm surprised at how MANY comments are popping up in this thread. clancy must have it a nerve! CS
oops
Oops, my previous comment was directed to McChris CS
If I learned this as a kid, i
If I learned this as a kid, it would probably be easier. But at this point I would probably forget to add the one to the left number. haha.
I have a confession to make.
I was just going to say "I'm not good at math and I can't do it in my head, so I've always carried some form of calculator around with me nearly everywhere I go."
But the truth is, I can actually do math in my head, and if I try, I can do math fairly well. I JUST DON'T LIKE TO!!!!!!! I hate it. I hate adding and subtracting, multiplication & division. I hate it. I find it annoying. And I think it's a waste of time when there's something called a calculator! And considering calculators come in tiny pocket size, are on cell phones, on digital watches (I had one of those once too of course)... there's really no reason not to use one, unless you actually enjoy doing it in your head or on paper. HAHA.
But because I feel like I probably look like a geek when I pull out a calculator in public, I usually say to people that I'm not good at math and can't do it in my head, and I'll go as far as to not even try and just act frustrated - when in reality, I just really can't be bothered. HAHAHA!!
I do think the Austrian method is definitely a paper-saver, so I'm in favour of it.
After my grandmother died, we found drawers full of scrap paper she'd used for written mathematics. Why she kept her calculations, I do not know. She was not working on the theory of relativity, she was just balancing her budget. haha. But I remember my sister remarked, "She was like a mad mathemetician, look at this!" haha. She was very good at finances, probably because she lived through The Depression, she had an incentive then that carried through the rest of her life. But I often wondered - why didn't she use a calculator once they had been around for many years?
It perplexes me. To me it seems like washing clothes by hand when you have a washing machine. haha.
It is really quite natural
I use a modified form of it myself. It is really the most natural way to subtract. Plus, you get an estimate of the answer from the beginning, which is often what you need anyway. If you only need your answer to the nearest 50 (for instance), you only need to do about half the calculation.
I was never taught to subtract this way formally. Of course, being a mathematician I probably subtract more than the average person on a daily basis. :) It is good to see that this is actually taught formally somwhere.
It is strange to see that Americans learn to subtract from right to left when we learn to divide from left to right, which is really the same idea as the Austrian method of subtraction.