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Re: Please remind me why 3^2 = 9
Posted:
Nov 19, 2012 1:29 AM


If may well be that, within Beth, it is actually true that " 73^2 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 flowdiagrams. 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 "ab" actually *means* the same as "a+(neg b)" an unrealistic fallacy. Students enter beginning algebra with a wellentrenched 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 "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 belowzero 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 ab. Far better for all concerned to abide with students' entrenched, "takeaway" meaning of "subtraction" ... and then to lead them to *derive" the formulaequivalence theorem: a+(neg b) = ab.
[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 (shortcut) "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 nonassociativity of linear verbiage is by no means unique to mathematics. [Does "the light brown fox" mean "the lightbrown fox (not black or even dark brown)" ... or "the light brownfox (who is not very heavy)"] Some persons have much fun reassociating 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 lefttoright 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 flowdiagrams for formulas for functions can greatly strengthen their understandings 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 lefttoright, 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
> 73^2 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 > grammarrule 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 73^2, many students take the "" to mean subtraction. > Then their question becomes, "Does it mean (73)^2? [as 4^2] ... or > 7(3^2) [as 79]?" 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 "73" is 7+(1)*3. But would that be > commonsensible to students? > > # So the instructological question is whether or not your students > actually grasp and apply your theorem: neg of (a) = (1below0)*a. Even if > not, they still might use your symbolswitching 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 > syntaxconventions 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 > **************************************************************************** * To post to the list: email mathedcc@mathforum.org * * To unsubscribe, email the message "unsubscribe mathedcc" to majordomo@mathforum.org * * Archives at http://mathforum.org/kb/forum.jspa?forumID=184 * ****************************************************************************



