So, what is this unary operation of negation? If I write the number 3, have I written an operation? To me, I have not. I have written the symbol for a negative number. I understand it in the context of, say, ³square 3, then ³negate² the answer². The common interpretation would be 3^2 of course, but as I have said, the abstract version is, I believe, too ambiguous to safely use. I think an equally valid statement like ³write down a negative 3, then square it² could produce the same picture, at least to those who cannot remember the other interpretation.
It does occur to me that stressing the ³correct² interpretation of 3^2 with students develops no understanding of operations that I can see, and it does consume an amount of time and create more consternation than it is worth. If I only saw it once in a while in texts, I wouldn¹t have commented, but every text makes a big, special deal about this unholy sequence of symbols, and every teacher makes sure it¹s on the final. :-)
On 10/18/12 2:38 PM, "Guy Brandenburg" <firstname.lastname@example.org> wrote:
> Well, lets think of a nice way to remember it! > > How about this: > Please! escape! No! Medical Doctor! Ambulance Surgeon! > > Guy > > On Oct 18, 2012, at 14:24, "Collinge, Peter (Mathematics)" > <email@example.com> wrote: > >> Part of why students are confused about 3^2 is that our standard teaching >> about order of operations is incomplete. We don¹t generally specify where >> negation fits into the order of operations. In fact, PEMDAS (if not banned) >> should be taught as P.E.N.MD.AS, with negation inserted after exponentiation >> and before multiplication/division. Then it would be clear that in -3^2, the >> exponentiation operation is performed before the negation. (I realize that >> some would say that negation is just multiplying by -1, so would have the >> same precedence as any multiplication, but clearly many students don¹t see >> that.) But unfortunately P.E.N.MD.AS just doesn¹t make a nice, >> easy-to-remember mnemonic. >> >> >> Peter Collinge, Professor >> Department of Mathematics >> Monroe Community College, Rochester NY 14623 >> >> E-mail: firstname.lastname@example.org <BLOCKED::">mailto:email@example.com> >> <mailto:firstname.lastname@example.org> >> Voice: (585)292-2943 >> <image001.jpg> >> >> >> From: email@example.com [mailto:firstname.lastname@example.org] On >> Behalf Of Jack Rotman >> Sent: Thursday, October 18, 2012 12:39 PM >> To: Wayne Ford Mackey; Guy Brandenburg; email@example.com; Philip >> Mahler >> Cc: mathedcc >> Subject: RE: Please remind me why -3^2 = -9 >> >> Phil and all: >> >> The expression ³-3^2² deals with the order of operations; the most advanced >> operations are always done first unless a grouping symbol forces¹ a lower >> prior operation to be done first. Since exponentiation is more advanced >> than the sign of a number, the exponent only applies to one symbol (the 3) >> when there are no grouping symbols. >> >> Instead of banning this type of problem, I believe that we should ban >> ³PEMDAS² or anything like it. The use of overly simplistic rules (often >> stated as a sequence of nouns) discourages learning and encourages >> memorization. If a large rate of correct answers is the only criteria, just >> have students use a calculator and train them on use of parentheses. If we >> are teaching mathematics, we should focus on understanding the priority of >> operations. What Phil internalized was this understanding; saying PEMDAS¹ >> does not provide any of the understanding to our students [I normally spend >> about a tenth of my time in class trying to undo the damage of PEMDAS. >> Undoing partially correct information is terribly difficult!] >> >> Even if we never showed ³-3^2², students would still be evaluating x^2 for >> x=-3; knowing that this means squaring a negative is a part of basic literacy >> in mathematics. >> >> For those with an interest, I¹ve posted some ³anti-PEMDAS² comments on my >> blog (www.devmathrevival.net <http://www.devmathrevival.net> ). You can use >> the search box on the site to find them. >> Jack Rotman >> Professor, Mathematics Department >> Lansing Community College >> (517)483-1079 firstname.lastname@example.org <mailto:email@example.com> >> www.devmathrevival.net <http://www.devmathrevival.net> >> >> >> From: firstname.lastname@example.org [mailto:email@example.com] On >> Behalf Of Wayne Ford Mackey >> Sent: Thursday, October 18, 2012 12:11 PM >> To: Guy Brandenburg; firstname.lastname@example.org; Philip Mahler >> Cc: mathedcc >> Subject: RE: Please remind me why -3^2 = -9 >> >> >> It should be read as the opposite of 3 squared. Since 3 squared is 9, the >> opposite is -9. The "-" sign is used in 3 different ways. In front of a >> natural number it means negative or minus, in front of anything else it means >> opposite and between two things it means add the opposite. >> >> >> >> wayne >> >> >> >> >> From: email@example.com [firstname.lastname@example.org] on behalf >> of Guy Brandenburg [email@example.com] >> Sent: Thursday, October 18, 2012 6:04 AM >> To: firstname.lastname@example.org; Philip Mahler >> Cc: mathedcc >> Subject: Re: Please remind me why -3^2 = -9 >> >> It's a convention. In a case like that, one really ought to use parentheses >> to make the meaning clear, since a lot of people, not just youngsters, will >> get confused. >> >> >> >> If one intends to say (-3)*(-3), then write (-3)^2. If one means - (3)*(3), >> then write - (3^2). >> >> >> >> Writing -3^2 is simply asking for confusion. >> >> >> >> Guy Brandenburg, Washington, DC >> http://gfbrandenburg.wordpress.com/ >> http://home.earthlink.net/~gfbranden/GFB_Home_Page.html >> ============================ >> >> >> From: "email@example.com" <firstname.lastname@example.org> >> To: Philip Mahler <email@example.com> >> Cc: mathedcc <firstname.lastname@example.org> >> Sent: Thursday, October 18, 2012 6:05 AM >> Subject: Re: Please remind me why -3^2 = -9 >> >> >> Phil, >> >> -3 means -1 x 3, so -3^2 is (-1)(3^2) = (-1)(9) = -9. >> >> John Peterson >> >> >> From: "Philip Mahler" <email@example.com> >> To: "mathedcc" <firstname.lastname@example.org> >> Sent: Thursday, October 18, 2012 5:37:07 AM >> Subject: Please remind me why -3^2 = -9 >> >> I have been teaching a long time, and I know from experience that 50% of >> students will tell me that 3^2 = +9 on a test or a final, despite having >> discussed it a few times in a course. >> >> When I first started teaching I taught calculus and precalc. Piece of cake. >> Then I started with an Algebra I class and couldn¹t connect at all for the >> first week or so. I was ready to believe I couldn¹t teach. I simply could not >> explain how I got the right answers when evaluating expressions... Then I >> discovered the order of operations (PEMDAS to some). A definition of the >> order of operations which I had so internalized that I didn¹t know there was >> a rule for it. Since that discovery I¹ve been a wonderful teacher. :-) >> >> So... I must be missing something that so many of my students think 3^2 is >> +9. What is the rule I have never discovered? >> >> Full disclosure: I think k^2, k a constant, should be banned from >> mathematics texts and tests. -x^2, x a variable, evaluated for say 3, >> absolutely (no pun intended) but not 3^2. >> >> Phil >> >