Arithmetic is the art of processing numbers.
We have ADD, SUBTRACT, MULTPLY and DIVIDE
In ordinary language these words are verbs which have a direct object and an indirect object.
“Add the OIL to the EGG YOLKS one drop at a time”.
“To find the net return subtract the COSTS from the GROSS INCOME”.
In math things have got confused.
We can say “add 3 to 4″or we can say “add 3 and 4”.
We can say “multiply 3 by 4” or we can say “multiply 3 and 4”.
At least we don’t have that choice with subtract or divide.
The direct + indirect form actually means something with the words used,
but when I see “add 3 and 4” my little brain says “add to what?”.
There are perfectly good ways of saying “add, or multiply, 3 and 4” which do not force meanings and usages onto words that never asked for them.
“Find the sum of 3 and 4” and “Find the product of 3 and 4” are using the correct mathematical words, which have moved on from “add” and “multiply”, and incorporate the two commutative laws.
If we were to view operations with numbers as actions, so that an operation such as “add” has a number attached to it, eg “add 7”, then meaningful arithmetical statements can be made, like
“start with 3 and then add 5 and then add 8 and then subtract 4 and then add 1”
which with the introduction of the symbols “+” and “-“, used as in the statement above allows the symbolic expression 3+5+8-4+1 to have a completely unambiguous meaning. It uses the “evaluate from left to right” convention of algebra, and does not rely on any notion of “binary operation” or “properties of operations”.
If we want to view “+” as a binary operation, with two inputs then, yes, we can ascribe meaning to “3+4”, but not in horrors such as the following (found in the CCSSM document):
To add 2 + 6 + 4, the second two numbers can be added to make a ten,
so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)
If + is a binary operation, which are the two inputs for the first occurrence of + and which are the inputs for the second occurrence of + ?
The combination of symbols 2 + 6 + 4 has NO MEANING in the world of binary operations.
See A. N. Whitehead in “Introduction to Mathematics” 1911.
here are the relevant pages:
And here are two more delights from the CCSSM document
subtract 10 – 8
add 3/10 + 4/100 = 34/100
In addition I would happily replace the term “algebraic thinking” in grades 1-5 by”muddled thinking”.