The teaching of subtraction in schools Subtraction

1 teaching of subtraction in schools

1.1 in america
1.2 in europe
1.3 comparing 2 main methods

the teaching of subtraction in schools

methods used teach subtraction elementary school vary country country, , within country, different methods in fashion @ different times. in is, in united states, called traditional mathematics, specific process taught students @ end of 1st year or during 2nd year use multi-digit whole numbers, , extended in either fourth or fifth grade include decimal representations of fractional numbers.

in america

almost american schools teach method of subtraction using borrowing or regrouping (the decomposition algorithm) , system of markings called crutches. although method of borrowing had been known , published in textbooks previously, use of crutches in american schools spread after william a. brownell published study claiming crutches beneficial students using method. system caught on rapidly, displacing other methods of subtraction in use in america @ time.

in europe

some european schools employ method of subtraction called austrian method, known additions method. there no borrowing in method. there crutches (markings aid memory), vary country.

comparing 2 main methods

both these methods break subtraction process of 1 digit subtractions place value. starting least significant digit, subtraction of subtrahend:

sj sj−1 ... s1

from minuend

mk mk−1 ... m1,

where each si , mi digit, proceeds writing down m1 − s1, m2 − s2, , forth, long si not exceed mi. otherwise, mi increased 10 , other digit modified correct increase. american method corrects attempting decrease minuend digit mi+1 1 (or continuing borrow leftwards until there non-zero digit borrow). european method corrects increasing subtrahend digit si+1 one.

example: 704 − 512.

−
1






c


d


u





7


0


4





5


1


2





1


9


2










⟵


c
a
r
r
y











⟵



m
i
n
u
e
n
d






⟵



s
u
b
t
r
a
h
e
n
d






⟵


r
e
s
t

o
r

d
i
f
f
e
r
e
n
c
e








{\displaystyle {\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 704, subtrahend 512. minuend digits m3 = 7, m2 = 0 , m1 = 4. subtrahend digits s3 = 5, s2 = 1 , s1 = 2. beginning @ 1 s place, 4 not less 2 difference 2 written down in result s 1 s place. in ten s place, 0 less 1, 0 increased 10, , difference 1, 9, written down in ten s place. american method corrects increase of ten reducing digit in minuend s hundreds place one. is, 7 struck through , replaced 6. subtraction proceeds in hundreds place, 6 not less 5, difference written down in result s hundred s place. done, result 192.

the austrian method not reduce 7 6. rather increases subtrahend hundred s digit one. small mark made near or below digit (depending on school). subtraction proceeds asking number when increased 1, , 5 added it, makes 7. answer 1, , written down in result s hundred s place.

there additional subtlety in student employs mental subtraction table in american method. austrian method encourages student mentally use addition table in reverse. in example above, rather adding 1 5, getting 6, , subtracting 7, student asked consider number, when increased 1, , 5 added it, makes 7.