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I have studied long ago different method for subtracting numbers, for instance the borrow method or the Austrian method (I hope I am using the right names; I am not an English native speaker and these do not correspond to the terminology used in french) and others whose names I don't know.

I would be interested in knowing how countries have decided what is the "main" method to teach in school, and if there is some kind of international study trying to determine whether some method provides better results than another.

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For the United States, I suggest as a resource:

Karp, K., Caldwell, J., Zbiek, R. M., & Bay-Williams, J. (2011). Developing Essential Understanding of Addition and Subtraction for Teaching Mathematics in Pre-K-Grade 2. National Council of Teachers of Mathematics. 1906 Association Drive, Reston, VA 20191-1502.

Here is a screenshot of page 45:

enter image description here

As you may notice from the image above, the book does cover the "Austrian method" about which you ask. You can also see in Reflect 1.14 at the top of the page that there is another method discussed called "partial differences."

The section on Properties and Algorithms [of Subtraction] (p. 35) includes a number of approaches and representations with respect to subtracting. Besides the Austrian Method and partial differences, there is also the so-called Standard Algorithm for Subtraction, as well as the use of additive inverses in what is called Compensation or Friendly Numbers (in the book, this is called the "same change algorithm"). In terms of presentations, these include number lines, hundreds charts, place-value blocks, descriptions in words, and symbolic representations.

For completeness, here is a picture of the book's table of contents:

enter image description here

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    $\begingroup$ Thanks a lot for this! Unfortunately it will be quite difficult for me to get this book, but I'll try. If you have other books or references I will be very pleased :) $\endgroup$ – user6828 May 31 '16 at 21:06
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    $\begingroup$ @Nekochan You can buy the book directly from NCTM's website either in print or (as I have it) PDF format. Here is a link (the price for the PDF is 23.06 USD). $\endgroup$ – Benjamin Dickman May 31 '16 at 22:06

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