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Re: Please remind me why -3^2 = -9
Posted:
Nov 19, 2012 1:29 AM
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If may well be that, within Beth, it is actually true that " 7-3^2 means
7+-3^2 means 7+-1*3^2" ,... or, alternatively, she may choose to "define"
any two of those formulas to mean the same as the third one. However, very
few students can internally accept that those three formulas have the same
*meaning* ... they simply do NOT have the same flow-diagrams. All three
formulas do generate the same function ... which gives a value of -2 to the
triad, (7,3,2). But achieving the equivalence of a function's various
formulas requires some derivations ... via theorems!
Throughout much of mathematics, conversions among equivalent formulas for
the same functions is a pervasive theme. Indeed, the reason for making a
thing of "the field properties" is that "a+b" does NOT have the same
*meaning* as "b+a" ... etc., etc.
In particular, the commonplace pedagogical belief that "a-b" actually
*means* the same as "a+(neg b)" an unrealistic fallacy. Students enter
beginning algebra with a well-entrenched meaning for "subtraction"... years
of experience with subtracting whole numbers and fractions. Even with
rationals, "taking away (13/5)Negs" raises the balance, but is NOT the same
kind of *transaction* as "adding on (13/5)Positives" ... as any banker or
credit card statement will verify.
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.
Pedagogically, that "definition" does not suffice with students who have
not yet personally concluded that a+(neg b) gives the same results as a-b.
Far better for all concerned to abide with students' entrenched,
"take-away" meaning of "subtraction" ... and then to lead them to *derive"
the formula-equivalence theorem: a+(neg b) = a-b.
[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." ]
But even that partial resolution of "-" does not answer Phil's question. The
condition that -3^2 *means* -(3^2) ... versus *meaning* (-3)^2 ... is an
"associative property" of the syntax for formulas ... neg(3pwr2) Vs.
(neg3)pwr2. The non-associativity of linear verbiage is by no means unique
to mathematics. [Does "the light brown fox" mean "the light-brown fox (not
black or even dark brown)" ... or "the light brown-fox (who is not very
heavy)"] Some persons have much fun re-associating other persons' remarks.
However, to even *attempt* to "prove it" ignores students' needs to learn
the language of formulas for functions. They MUST learn where the
(conventionally "understood") left and right parens are located. That does
not discredit Beth's equation ... as a mnemonic theorem for assisting with
calculations ... only that no two of her three formulas *mean* the same
composition of component functions.
Schremmer's point is well worth pursuing: "More generally, the trouble is
that we write a lot of shorthand without much delving on the longhand and
dealing explicitly with the default rules." The use of left and right
parens is but a left-to-right syntactic mechanism for expressing particular
modes of composing functions. In much the same way as students learn to
diagrams sentences or computer programs, their experiences in generating
flow-diagrams for formulas for functions can greatly strengthen their
under-standings of the linguistic conventions.
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.
I for one would be interested in knowing a reasonable *theory* about
how/why the conventions arose ... presumably from some interfacing of
left-to-right, linguistic syntax and mathematical semantics. Perhaps, even
more important than historical accuracy would be the formulation of a
palatable *legend* which would respond to Schremmer's challenge.
Cordially,
Clyde
- --------------------------------------------------
From: "Beth Hentges" <Beth.Hentges@century.edu>
Sent: Sunday, November 18, 2012 1:48 AM
To: "Clyde Greeno" <greeno@malei.org>
Cc: "Wayne Ford Mackey" <wmackey@uark.edu>; "Alain Schremmer"
<schremmer.alain@gmail.com>; "Philip Mahler" <mahlerp@middlesex.mass.edu>;
<mathedcc@mathforum.org>; "Clyde Greeno @ MALEI" <greeno@malei.org>
Subject: Re: Please remind me why -3^2 = -9
> 7-3^2 means 7+-3^2 means 7+-1*3^2
>
> The point is that it is all convention, there are reasons for choosing the
> conventions we do.
>
> I am one who usually uses the "flying hyphen" to indicate a negative or
> opposite sign.
>
> I have seen one elementary curriculum that used a hat to indicate negative
> numbers.
>
> Beth
>
>
> On Nov 15, 2012, at 1:36 AM, "Clyde Greeno" <clydegreeno@cox.net> wrote:
>
> Beth, thanks for responding!
>
> My #s are below
>
> --------------------------------------------------
> From: "Beth Hentges" <Beth.Hentges@century.edu>
> Sent: Wednesday, November 14, 2012 10:15 PM
> To: "Clyde Greeno" <greeno@malei.org>
> Cc: "Wayne Ford Mackey" <wmackey@uark.edu>; "Alain Schremmer"
> <schremmer.alain@gmail.com>; "Philip Mahler" <mahlerp@middlesex.mass.edu>;
> <mathedcc@mathforum.org>; "Clyde Greeno @ MALEI" <greeno@malei.org>
> Subject: Re: Please remind me why -3^2 = -9
>
>> (-3)^2 is clear
>>
>> -(3^2) is clear
>>
>> We have to choose which we mean when we write the following.
>>
>> -3^2
>> # So far, so good!
>
>
>> The reason we choose -3^2 to mean -(3^2) is because we do powers before
>> multiplication and because -a = -1*a.
> # And the reasons for that rule and for that equation are ????
> It seems as though you are trying to *conclude* what is actually a
> grammar-rule of the written language. The "powers before multiplication"
> is just such a grammar rule, and it gains no rational support from the
> equation. Nonetheless, I can believe that your use of that equation very
> well might help train students to calculate, better ... even without
> grasping the commonsensibility of it all.
>
> # The mathematical issue with your equation is that it does the usual
> curricular "slight of hand" by ambiguously using "-" with two differing
> meanings ... while many students use even a third meaning. Within the
> context of linear number systems. "-1" means "1 below 0" ... while "-a"
> means "the negative/opposite of a" ... while students routinely take "-"
> to mean "subtract a."
>
> # In the case of 7-3^2, many students take the "-" to mean subtraction.
> Then their question becomes, "Does it mean (7-3)^2? [as 4^2] ... or
> 7-(3^2) [as 7-9]?" Would your equation lead some to interpret it as
> 7(-1)*(3^2)? Of course an instructor might simply *define" subtraction
> so that the meaning of "7-3" is 7+(-1)*3. But would that be
> common-sensible to students?
>
> # So the instructological question is whether or not your students
> actually grasp and apply your theorem: neg of (a) = (1-below-0)*a. Even if
> not, they still might use your symbol-switching device for more
> systematically calculating, correctly.
>
> # But as yet, I fail to perceive how your equation makes algebraic
> formulas any more commonsensible than would a good treatment (much better
> than usual) of how to use parentheses in accord with the
> syntax-conventions that currently are in commonplace use throughout the
> field (and in most calculators).
>
> Cordially,
> Clyde
>
>
>>
>> (-3)^2 = (-1*3)^2 , and the parentheses tell us to do the multiplication
>> first.
>>
>> -(3^2) = -1*(3^2), and the parentheses tell us to do the power first.
>>
>> -3^2 = -1*3^2, and we do powers before multiplication.
>>
>>
>> As for PEMDAS, I use PEMA. Otherwise, students think PEMDAS says to do
>> multiplication before division. Also, when I write it in words, for the
>> E for exponents (which really should be another P for powers), I write,
>> "Do exponents and roots from left to right." So, even if I did use
>> PEMDAS it would be PERMDAS. Then, we also have to be careful with
>> absolute value as well.
>>
>> Beth in MN
>
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