On Oct 24, 2012, at 12:06 AM, Paul Tanner <firstname.lastname@example.org> wrote:
> The teaching and learning of arithmetic formulas (especially involving > fractions) and their applications is not algebra foolishness.
I don't consider the teaching and learning of formulas to be algebra at all. Just because it has letters in it doesn't make it algebra. Not until the student treats those letters as numbers (rather than blanks to fill in) and starts doing arithmetic with them (on their own) do I consider it algebra. When you describe the area of a rectangle as "length x width" or "L x W" that isn't algebra. It isn't even the beginning of algebra. It is the beginning of making formulas.
Algebra begins when the length of the rectangle is composed of two segments, a and b, and the width of the rectangle is composed of two segments, c and d, and the student, on their own, writes area = (a + b) x (c + d). In other words, the student has transcended the urge to ask "What numbers do we use for a, b, c and d?"
> > The teaching or learning of methods or techniques like being able to > solve for x in such as > > 53/100 = 37/x > > in just one step is not algebra foolishness. (Equal fractions with one > of the denominators being 100 is a standard set-up in percent
You meant to say "pretend to solve for x", right? Or are you pretending to us that you just taught them to "solve"? This is my point. When you are walking through the steps to "solve" for x, the only thing the students can do is nod their heads. If they actually followed your "solving" then they would go forth and start solving a bunch of stuff.
I am not against formulas for fraction arithmetic, but they must be developed arithmetically, not algebraically. In your example above, the student must "simply see" when and where to multiply or divide. I recognized this early on with my son and word problems...
Joe has 20 apples and Mary has twice as many. Joe has 20 apples which is half as many as Mary has. Mary has 40 apples and Joe has half as many. And so on...
These problems are a level below your ratio problem, but when I was first going through these with my son (over a year ago) I had the very real urge to break them down (algebraically?) and bring order to the permutations. Part of that urge stems from the fact that I have already been down the path (I am algebraic) and part of the urge stems from the frustration of watching him fumble. But I didn't. I hung with it, if you call a month hanging with it. He came out the other end of that month with "the sense". And what I learned from all that is to be patient and let the math do its magic.