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Re: Please remind me why 3^2 = 9
Posted:
Nov 19, 2012 8:58 AM


On Nov 19, 2012, at 1:29 AM, Clyde Greeno @ MALEI wrote:
> To "define" the expression "ab" to mean the formula, "a+(neg b)" > is merely a simplistic expedience ... for use among persons for whom > such a shortcut definition suffices. Mathematically, it does NOT > suffice for subtracting within linear systems which have no below > zero numbers: whole numbers, fractions, or decimals.
Part of the problem is that once teachers have entered mathematics, they forget the real world that mathematics represents and then they glide from "plain" (=unsigned) numbers to signed numbers.
What is important when dealing with beginners is to let them realize that we must recycle symbols/terms so that their meaning is context dependent. How many textbooks do you know that deal with that issue other than in one paragraphif that?
For instance, how many books do you know that heed Greeno's "[The students] MUST learn where the (conventionally "understood") left and right parens are located." by first using parens and then deciding which can be omittedat the cost of "default rules".
For another instance, there is addition for plain numbers and there is addition for signed numbers and to define the latter requires comparison, addition and subtraction of plain numbers; immediately recycling symbols then makes matters near incomprehensible. But how many books do you know that use different symbols for the two? (This is why, for integers, I start by using oplus and ominus.)
> Sure, students can earlier be trained to mimic teachers ... and > teachers can kid themselves into thinking that students "understand" > such formal trickery. But until the student internally owns both > concepts ... and also owns that both processes produce the same > result ... there cannot be an actual understanding of that (short > cut) "definition."
Which is why the exclusive focus on computations is a disaster. What should be presented to the students is the need to represent the real world on paper and the many difficulties thereof. Computing is only one aspect of this inasmuch as computation represent realworld processes. But, at the same time, we also need to make statements about the real world. Etc.
> So arise their natural questions about WHY those particular > syntactic conventions ... such as 3^2 meaning neg(3pwr2) Vs. > (neg3)pwr2. Beth dutifully tries to answer such questions, > mathematically. They cannot be answered by pure logic, but the > "rules" are not "arbitrary" in the sense of being willfully > dictated. Rather, they have evolved, over time, as matters of > expedience for those who published mathematics (long before the days > of mass schooling). There were *reasons* for that evolution.
There certainly must have been reasons but these answered specific needs of the particular community that adopted them. For instance, f(x) is adapted to analysis while the "reverse Polish" xf is adapted to abstract algebra. Similarly, as Matthews pointed out, the rules under discussion here, pace EddieC, are adapted to polynomials.
It is worth realizing that there CANNOT be a best notation. For instance, algebraist need to "evaluate" functions, for instance they need to compare xf and xg. But then x operates as an "evaluation function" ... written in "straight Polish".
Regards schremmer
P. S. I am arguing from the platonist (aka model theoretic) viewpoint. If you happen to be a formalist, the argumentation is different but the linguistic issues remain.
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