
Re: Is this an exceptionally hard set of questions to answer?
Posted:
Oct 29, 2002 12:38 PM


Alberto Moreira <junkmail@moreira.mv.com> writes:
> I would postpone the concept of > associativity to senior high or so, when they first need to face it in > its formal embodiment.
Associativity (and commutativity and distributivity) are needed and used much sooner than senior high. For example, distributivity is used extensively in the "long multiplication" algorithm.
> The fact that 6=5+1=1+2+3 doesn't need to be explained to a kid, it > evolves naturally as the kid's number intuition develops
There's a little trap there: "1+2+3" does not even make sense until you know that addition is associative.
The trouble with intuition is that it easily can be, and often is, wrong. Indeed, it is intuitive to many students that (1/2) + (2/3) = (1+2)/(2+3). The student who understands the properties of the operators is (in my experience) less likely to make this sort of error, while the student who merely develops an intuition that operations "behave nicely", without understanding exactly what "nicely" means, expects that the addition of fractions can be accomplished using the "nice" and "intuitive" method.
I have heard comments that arithmetic and basic algebra appear as "magic" to students, that there is some mysterious force involved that makes things work. Things are only magic if you don't know what's going on, if you don't know the rules.
> We have fingers  our hands are our abacuses, and couting with fingers > is intuitive. An abacus is just an extension of our hands.
Nonsense. With two rows on an abacus, you can count to 100. With two pairs of hands, you can count to 20 (assuming you use the usual simple form of fingercounting). If you really think that an abacus is an extension of our use of our fingers as counting tools, you don't understand our digits representation of numbers (or you don't know how to use an abacus).
> And as I pointed out before, the length of a rod is a > false representation of a number: I can call the same length "1", "2" > or "3.141592...", depending on how and where I use it. But a digit is > a digit, and a finger is a finger.
If you refuse to recognize that the length of a rod is a perfectly valid representation of a number once you have agreed on what has length 1, fine. Make little cuts on the rods so that it looks like a bunch of cubes glued together, then you can count the cubes. Many children have no difficulty imagining these cuts (or similarly imagining the rod made up of cubes). Apparently you do.
Kevin.

