
Re: Is this an exceptionally hard set of questions to answer?
Posted:
Oct 31, 2002 9:06 AM


Kevin Foltinek <foltinek@math.utexas.edu> said:
>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.
You don't need to bother with the property, which is merely a formalization of the intuition. At this level, the intuition of addition, through counting, is enough. Long division is a shorthand for repeated subtraction, which, again, is intuitive. For example, if you divide 7321 by 16 using the algorithm, you single out "73" which is actually 7300. Dividing 73 by 64 gives you 4x16=64 and remainder 9, that's actually 6400 + 900. What you are actually doing is subtracting 16 hundreds from 7321 4 times:
73211600160016001600 = 921
The associativity you need to worry about operates at an intuitive level, no need to bother about formalizing it. If a student has number sense, it's jolly easy to point out what the algorithm is doing, but if he or she doesn't, it's nearly impossible to justify it, associativity or not. >There's a little trap there: "1+2+3" does not even make sense until >you know that addition is associative.
In intuitive terms, 1+2+3 makes all the sense in the world: count one, than count two more, than count three more. Counting to fifteen is nothing but adding 1+1+1+1+1+1+1+1+1+1+1+1 +1+1+1. The fact that later on mathematics models addition as a function that takes two arguments and barfs out a result is a consequence of the mathematical MODEL of addition, not of the intuitive concept of counting or adding.
So, I come back to my point: before the restricted model is taught, we must sediment the intuition.
>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).
It is not intuitive, and it's a consequence of the perversity of the archaic notation we use to represent arithmetic. If you write (1/2)+(2/3), you are saying "divide 1 by 2, take the result, put in your pocket; divide 2/3, and add the result to the value you put inside your pocket. That is what the NOTATION is saying, it has nothing to do with the intuition of the operations themselves. I could put it in postfix notation and state it as "1 2 / 2 3 / +", or better, something like "1 2 / @a 2 3 / @b a b +" and that would possibly capture the intent of the notation way better: take 1, take 2, divide, put the result in a, take 2, take 3, divide, put the result in b, take the value in a, take the value in b, add. No doubt whatsoever about what's the sequence the operations will be performed.
The point is, we use a CONVENTION that division has higher precedence than addition. So, 1/2+3 means (1/2)+3 and not 1/(2+3), and that's a mere notational convention. The intuition is, what the sequence of the operations ? If I divide before I add, I get one result; if I add before I divide, I get another result. The notation establishes a sequence, but notational issues are about the notation, and not about the semantics of the operations themselves. Now, if I have a solid number sense, I will note that 1/2+3/4 can be ambiguous, unless we establish the notion of precedence among operations. And I do believe that confusing notation with concept is a serious problem that many teachers of elementary mathematics don't seem to be able to rid themselves of.
>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.
The student who understands the way the operations themselves work will have little problem fathoming the ins and outs of notation. The issue here isn't the understanding of how the operations work, but the understanding of what the notation is trying to say. Two different issues !
>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.
Arithmetic is intuitive. Arithmetic notation is contrived. There are two problems here, one is to understand arithmetic, the other is to grasp our notation of arithmetic. We mustn't confuse the two issues ! >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).
Counting in tens is enough for elementary school. For example, say we want to add 74+28. I count 8,9101112, so, I jot down 2, and carry one. Now I count, 7, 89 (for the tens), 10 (for the carry), and I jot down the 10 next to the 2: result, 102. Man, I used that when I learned arithmetic with my father even before I went to school. I still remember my little first and second grade colleagues counting in their fingers to figure out the result of an addition, because that was intuitive enough  and my teachers would bother them nonstop to get them to stop using their fingers instead of leveraging on that intuitive impulse.
The important thing is to develop the concept of carry, and do things one column at a time. >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.
It is obvious, to me at least, that digital feel is a lot more exact than analog feel. So, length is not a good thing to use, unit is. Don't glue anything, just count  and here, cubes or rods are as good as one's own fingers. The difference is, our fingers are at the tips of our hands all the time.
And if nothing else, in this age of computers, digital intuition is a whole lot more useful than its analog counterpart.
Alberto.

