On Wed, Mar 14, 2012 at 2:03 PM, Paul Tanner <firstname.lastname@example.org> wrote:
<< SNIP >>
> > Do you really think that the relationships (ab)/b = a and (a/b)b = a > where the given quotients are defined is beyond an early elementary > school kid's capabilities? I don't think so, and the world-beating > South Koreans agree: Don't you recall what I shared before about how > they introduce subtraction and division, that they introduce them as > the inverses of addition and multiplication, where via this inverse > relationship one operation "undoes" the other? This is an algebraic > way of thinking, actually. They also use the number line from the > beginning as an aid. I think this also is algebraic as well as > geometric, since it immediately instills a mode of thinking, a sense > of scale, which has some algebraic qualities to it. >
Yes agree. Not really a new topic with us right.
In the curriculum I teach, one of the exercises is to use the multiplication operator to stand for composition.
You pass a function to a class named Composable and get back an object that will compose with another composable via the multiplication operator *.
The does / undoes relationship of inverse functions is clear here.
If the function is one that reverses a string, then to reverse the reverse is to do nothing, take the null action (identity action). So f * f = 1. Other functions behave differently obviously.
In our syntax, exponentiation is ** and the student is responsible for taking the Composable class above (already presented in the text) and making it such that f ** 2 = f * f such that (f * f)(x) == f ( f ( x ) ).
In the case of Rational Number objects, the inverse of any number is its reciprocal and clearly to divide is to multiply by the inverse, just as in any group. Nothing surprising there.
We come back through spiraling, to good ol' Fractions again. De nada. Identity is 1 (for multiplication), 0 for addition. We've been looking at permutations, along with integers modulo N.
Clearly this is more than we would usually tackle before secondary school and indeed this might be some 17 year old's reading, I have no real control over the age of my students.