This algorithm is used in Europe, and if you have students who first learned how to subtract in Europe, they may be using this method. The difference is that, if you have to borrow or regroup, instead of decreasing the digit in the next place on top by one, you increase the digit in the next place on the bottom by one. In our example
instead of changing the 8 to a 7, you change the 5 to a 6. In both methods you change the 3 to a 13. the way this is accomplished notationally is to put the 1 that you would use to make the 13 between the 3 and the 5.
The little one serves two purposes. It tells you that you are subtracting 7 from 13, and you can think of it as being added to the 5 to make a 6 which you subtract from the 8
The justification for the Austrian method is exactly the same as the justification for the standard algorithm. If we look at the picture we used to justify the standard algorithm,
the one ten that we regrouped into 10 ones is an extra ten that has been taken away. As a result, if you count the number of tens that are x'ed out, you will see that there are a total of 6 big x's.
People who use the Austrian method are impressed by how much neater it is than the standard algorithm, particularly if you have a long problem. If you consider
the standard algorithm looks like
which 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. With the Austrian method the problem looks like
and you subtract 8 from 12 to get 4 in the one's place, 7 from 13 to get 6 in the ten's place, 6 from 11 to get 5 in the hundred's place, and 5 from 7 to get 2 in the thousand's place.