Thank you Barry for yet another valuable article. I am an old chalkie and hope to pen you a couple of views of my own along the lines of "traditional" education. You may delete them if they offend.
A tangential thought: while you can't use associativity with subtraction as ordinarily expressed the rule works perfectly fine when combined in the right way with the distributive law and regarding subtraction as a combination of multiplication and addition:
9-5-3 = ?
On the one hand (9-5)-3 = 4-3=1 whereas 9-(5-3) = 9-2 = 7.
Probably a TERRIBLE thing to get middle-schoolers to muddle through though while they're first mastering the rules. It has its place when simplifying algebra, though. Eventually one learns to fluently convert to sums as needed. And in compound quotients sometimes to convert to products.
Thank you Barry for yet another valuable article. I am an old chalkie and hope to pen you a couple of views of my own along the lines of "traditional" education. You may delete them if they offend.
Well done Barry.
A tangential thought: while you can't use associativity with subtraction as ordinarily expressed the rule works perfectly fine when combined in the right way with the distributive law and regarding subtraction as a combination of multiplication and addition:
9-5-3 = ?
On the one hand (9-5)-3 = 4-3=1 whereas 9-(5-3) = 9-2 = 7.
But (9-5)-3 = (9+(-1)5)+(-1)3 = 9+((-1)5+(-1)3) = 9+(-1)(5+3) = 9-(5+3) = 1
whereas 9-(5-3) = 9+(-1)(5+(-1)3) = 9+(-1)5+(-1)(-1)3 = (9+(-1)5)+3 = 7
Probably a TERRIBLE thing to get middle-schoolers to muddle through though while they're first mastering the rules. It has its place when simplifying algebra, though. Eventually one learns to fluently convert to sums as needed. And in compound quotients sometimes to convert to products.