On Nov 9, 2012, at 12:50 PM, Joe Niederberger <firstname.lastname@example.org> wrote:
> I'd like to combine this with some observations about variables and language. In a prior post Robert Hansen question whether children can know the "significance" of variables like "x". I didn't quite know what to make of that position, but I'll guess he meant that they are not ready for contemplating algebraic expressions as entities in their own right. If so I agree, but I also agree with Clyde that they certainly can understand "length x width".
Maybe a picture is worth a thousand words. Attached is a photo of the whiteboard after last night's session with my 9 year old son. Understand that there were two boards full (erased) before this one. We were discussing area and perimeter and deriving formulas using letters. Some will say "Wow, that looks like algebra." but it isn't actually algebra. It is arithmetic with letters. It's a damn good precursor but it ain't algebra, yet. On a side note, he got that "ab" is the same as a x b and that you can't write 2 x 2 as "22" and with regards to multiplicands and multipliers, for a square he first wrote "a4" as the perimeter (he was really getting into not writing the times symbol) and I told him that you normally write the number first and the letter second. He said "Yeah, a4 is like saying "a 4's" and I said "Yep." Later I will explain to him that there is nothing wrong with "a 4's" so that he doesn't get foiled by (a + 4)(a + 4). But I appreciated that he saw sense in "standard form". He also understands exponents, as shorthand for repeated multiplication. Don't tell Devlin.
If this is what you guys are leading to with vectors and bones, then pardon me. We all seem to be in agreement.