
Re: How would YOU simplify arithmetic?
Posted:
Jun 29, 2012 1:23 PM


> 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 KIntroAlg are mathematical theories for calculating with numerals ... which represent various kinds of number systems. [Not to be confused with numbertheory]. Shifting from baseten numerals to base8 numerals not only changes the numeralsvocabulary, it also changes the calculationstheorems ... and so changes the arithmetic theory. The change from Arabicnumerals arithmetic to the commonplace arithmetics for fractions "confuses 5 kids out of every 4". The change to decimalpoint 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 numeralsvocabulary, but perhaps by using the same vocabulary. [Example: the prevailing addfromtheright theorem about calculating with Arabic numerals might be replaced by an addfromtheleft theorem ... not just changing the pedagogy, but changing the mathematical content of the calculationstheory taught to the child.] Of course, changing a mathematical theory of numeralscalculations manifests also as changes in WHAT is taught ... but perhaps not in HOW it is taught.
2) The pedagogy of numeralscalculations 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 binarynumerals] might be equally unwise. But the two kinds of "simplify" are quite different ... and they require very different kinds of instructological research.
Cordially, Clyde
  From: "Joe Niederberger" <niederberger@comcast.net> Sent: Friday, June 29, 2012 11:01 AM To: <mathteach@mathforum.org> 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

