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Re: Please remind me why -3^2 = -9
Posted:
Nov 19, 2012 8:58 AM
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On Nov 19, 2012, at 1:29 AM, Clyde Greeno @ MALEI wrote:
> To "define" the expression "a-b" to mean the formula, "a+(neg b)"
> is merely a simplistic expedience ... for use among persons for whom
> such a short-cut 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 paragraph---if 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 omitted---at 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 under-standing 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 real-world
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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