> I still don't understand - you want to simplify arithmetic *itself*, not > the *teaching* of it?
Two very different things.
1) The arithmetics that are currently taught K-IntroAlg are mathematical theories for calculating with numerals ... which represent various kinds of number systems. [Not to be confused with number-theory]. Shifting from base-ten numerals to base-8 numerals not only changes the numerals-vocabulary, it also changes the calculations-theorems ... and so changes the arithmetic theory. The change from Arabic-numerals arithmetic to the commonplace arithmetics for fractions "confuses 5 kids out of every 4". The change to decimal-point numerals also changes the arithmetic theory.
The Crabtree question is of how to change the mathematical theory of arithmetic (as currently taught in schools) ... perhaps by also changing the numerals-vocabulary, but perhaps by using the same vocabulary. [Example: the prevailing add-from-the-right theorem about calculating with Arabic numerals might be replaced by an add-from-the-left theorem ... not just changing the pedagogy, but changing the mathematical content of the calculations-theory taught to the child.] Of course, changing a mathematical theory of numerals-calculations manifests also as changes in WHAT is taught ... but perhaps not in HOW it is taught.
2) The pedagogy of numerals-calculations arithmetic can be changed without changing the underlying mathematical theory. [Example: telling students how to calculate ... Vs ... guiding them to discover that [such and such] will always work.]
Simplifying the pedagogy might be disastrous. Simplifying the mathematical theory of arithmetic [say, changing everything to binary-numerals] might be equally unwise. But the two kinds of "simplify" are quite different ... and they require very different kinds of instructological research.
- -------------------------------------------------- From: "Joe Niederberger" <firstname.lastname@example.org> Sent: Friday, June 29, 2012 11:01 AM To: <email@example.com> Subject: Re: How would YOU simplify arithmetic?
> I still don't understand - you want to simplify arithmetic *itself*, not > the *teaching* of it? > > (I'm with R.H. here - it is what it is - you can't simplify it by decree. > I disagree with R.H when he says that it *is* simple already - so why are > there still number theorists?) > > That's why I asked how do you define simplicity. > > Joe N