Editing Subtraction (section)

==The teaching of subtraction in schools==
Methods used to teach subtraction to [[elementary school]] vary from country to country, and within a country, different methods are adopted at different times. In what is known in the United States as [[traditional mathematics]], a specific process is taught to students at the end of the 1st&nbsp;year (or during the 2nd&nbsp;year) for use with multi-digit whole numbers, and is extended in either the fourth or fifth grade to include [[decimal representation]]s of fractional numbers.

===In America===
Almost all American schools currently teach a method of subtraction using borrowing or regrouping (the decomposition algorithm) and a system of markings called crutches.<ref name="Klapper1916">{{cite book |author=Klapper |first=Paul |url=https://archive.org/details/teachingarithme00klapgoog |title=The Teaching of Arithmetic: A Manual for Teachers |year=1916 |pages=[https://archive.org/details/teachingarithme00klapgoog/page/n94 80]– |author-link=Paul Klapper |access-date=2016-03-11}}</ref><ref>Susan Ross and Mary Pratt-Cotter. 2000. "Subtraction in the United States:  An Historical Perspective," ''The Mathematics Educator'' 8(1):4–11. p. 8: "This new version of the decomposition algorithm [i.e., using Brownell's crutch] has so completely dominated the field that it is rare to see any other algorithm used to teach subtraction today [in America]."</ref> Although a method of borrowing had been known and published in textbooks previously, the use of crutches in American schools spread after [[William A. Brownell]] published a study—claiming that crutches were beneficial to students using this method.<ref>{{cite journal |title=Subtraction From a Historical Perspective |journal=School Science and Mathematics |year=1999 |last1=Ross |first1=Susan C. |last2=Pratt-Cotter |first2=Mary |volume=99 |issue=7 |pages=389–93 |doi=10.1111/j.1949-8594.1999.tb17499.x }}</ref> This system caught on rapidly, displacing the other methods of subtraction in use in America at that time.

===In Europe===
Some European schools employ a method of subtraction called the Austrian method, also known as the additions method. There is no borrowing in this method. There are also crutches (markings to aid memory), which vary by country.<ref>Klapper 1916, pp. 177–.</ref><ref name="Smith1913">{{cite book |author=David Eugene Smith |title=The Teaching of Arithmetic |url=https://archive.org/details/bub_gb_A7NJAAAAIAAJ |year=1913 |publisher=Ginn |pages=[https://archive.org/details/bub_gb_A7NJAAAAIAAJ/page/n85 77]– |access-date=2016-03-11 }}</ref>

===Comparing the two main methods===
Both these methods break up the subtraction as a process of one digit subtractions by place value. Starting with a least significant digit, a subtraction of the subtrahend:
:''s''<sub>''j''</sub> ''s''<sub>''j''−1</sub> ... ''s''<sub>1</sub>
from the minuend
:''m''<sub>''k''</sub> ''m''<sub>''k''−1</sub> ... ''m''<sub>1</sub>,
where each ''s''<sub>''i''</sub> and ''m''<sub>''i''</sub> is a digit, proceeds by writing down {{nowrap|''m''<sub>1</sub> − ''s''<sub>1</sub>}}, {{nowrap|''m''<sub>2</sub> − ''s''<sub>2</sub>}}, and so forth, as long as ''s''<sub>''i''</sub> does not exceed ''m''<sub>''i''</sub>. Otherwise, ''m''<sub>''i''</sub> is increased by 10 and some other digit is modified to correct for this increase. The American method corrects by attempting to decrease the minuend digit ''m''<sub>''i''+1</sub> by one (or continuing the borrow leftwards until there is a non-zero digit from which to borrow). The European method corrects by increasing the subtrahend digit ''s''<sub>''i''+1</sub> by one.

'''Example:''' 704 − 512.

{{equation|
   \begin{array}{rrrr}
        & \color{Red}-1 \\
        & C & D & U  \\
        & 7 & 0 & 4 \\   
        & 5 & 1 & 2 \\
      \hline
        & 1 & 9 & 2 \\
   \end{array}
   \begin{array}{l}
       { \color{Red}\longleftarrow \rm carry }\\
       \\
       \longleftarrow \; \rm Minuend\\
       \longleftarrow \; \rm Subtrahend\\
       \longleftarrow \rm{Rest \; or \; Difference}\\
   \end{array}
}}

The minuend is 704, the subtrahend is 512. The minuend digits are {{nowrap|1=''m''<sub>3</sub> = 7}}, {{nowrap|1=''m''<sub>2</sub> = 0}} and {{nowrap|1=''m''<sub>1</sub> = 4}}. The subtrahend digits are {{nowrap|1=''s''<sub>3</sub> = 5}}, {{nowrap|1=''s''<sub>2</sub> = 1}} and {{nowrap|1=''s''<sub>1</sub> = 2}}. Beginning at the one's place, 4 is not less than 2 so the difference 2 is written down in the result's one's place. In the ten's place, 0 is less than 1, so the 0 is increased by 10, and the difference with 1, which is 9, is written down in the ten's place. The American method corrects for the increase of ten by reducing the digit in the minuend's hundreds place by one. That is, the 7 is struck through and replaced by a 6. The subtraction then proceeds in the hundreds place, where 6 is not less than 5, so the difference is written down in the result's hundred's place. We are now done, the result is 192.

The Austrian method does not reduce the 7 to 6. Rather it increases the subtrahend hundreds digit by one. A small mark is made near or below this digit (depending on the school). Then the subtraction proceeds by asking what number when increased by 1, and 5 is added to it, makes 7. The answer is 1, and is written down in the result's hundreds place.

There is an additional subtlety in that the student always employs a mental subtraction table in the American method. The Austrian method often encourages the student to mentally use the addition table in reverse. In the example above, rather than adding 1 to 5, getting 6, and subtracting that from 7, the student is asked to consider what number, when increased by 1, and 5 is added to it, makes 7.