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”.

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RE: “muddled thinking”

I get them to bet on what they call the “tidy” way — I know that there are infinitely many muddled ways — so things will always go toward muddle and mixedness.— Gregory Bateson,

Steps to an Ecology of Mind(1972)Hola Maya

I am trying to get his book from Amazon but their system is not working for me today.

So Bateson is the son of Margaret Mead. My parents had a copy of her famous book, “The Sexual Life of Savages”, which of course I had a look at!

G. Bateson and M. Mead were a couple; their daughter, though, is also an author: Mary Catherine Bateson. (Known best for her ideas around composing one’s own life story, which I would highly recommend!)

You can find an excerpt from M.C. Bateson in a

Life of Best Fit.You are perfectly right ! I failed to do “close reading” on the bit at the bottom of the Google Books excerpt.

Hola ! Bateson’s Steps to an Ecology of Mind arrived yesterday. I can’t get enough of it. The talks with Daughter are brilliant and very funny, witty and quite brutal towards the “conventional way of doing things”. I skipped a chapter and am now on “Art…”. Wow ! How come I never heard of this guy before?

Thanks a million, or even “Thanks a billion”.

From these examples you show on your blog I conclude that much emphasis in math education today is put on translating back and forth between plain English and math. I had always figured you should learn math as you learn another language – trying to think in this language as soon as possible, instead of translating all the time.

Being able to switch from the natural language form to the symbolic form is absolutely vital for any sort of real success in maths (I am excluding the rote learning and mindless application of stuff, even when it gets you the “right answer”.

My point is that the symbolic stuff is introduced in a gobbledegook fashion, where 3+4 has several language equivalents. Are we treating + as a binary operation? Sometimes. Are we treating it as an instruction to add 4 to 3?. Sometimes. Are we treating it as a way of seeing the number 7 ? Sometimes. If the kids get confused and resort to “Do what the teacher says” then the future for them is mathematically bleak.