THE tragic failure of mathematics education in the United States is the absence of referents for the symbols that get shoved around by the seemingly mindless rules. A more concise way to say this is mathematics is taught as though the symbols refer to themselves rather than to quantities or processes. The Mathematician will scoff at me while the Physicist will applaud. In Physics, we deal with quantities; whereas in Mathematics, we deal with … uh … hmmm … rules and symbols! Yes! Rules and symbols.
I could drone on in writing here; but I think that it will be more efficient if you watch this quick little video that I made that proves that numbers do not add. Enjoy.
Clark Vangilder's Cognitions 
Seems to me you are only saying you can’t add apples and oranges! I don’t think that idea is new or novel.
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If only it were that simple Joe! Of course you can “add” apples and oranges so long as you convert them to fruit. If you want to add things then either you add homogenous quantities as they come, or you change them into a common quantity.
What I illustrate here demonstrates how important the symbol-referent connection is. If one is not careful, then they extend that subtle assumption that we are adding things of the same kind.
The main point is that quantities add, whereas numbers do not … unless of course you assume that numbers are always parts of the same whole–which is not a valid assumption.
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