The expression "-3^2" deals with the order of operations; the most advanced operations are always done first unless a grouping symbol 'forces' 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 "PEMDAS" 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 'PEMDAS' 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 "-3^2", 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, I've posted some "anti-PEMDAS" comments on my blog (www.devmathrevival.net ). You can use the search box on the site to find them.
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.
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).
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
-3 means -1 x 3, so -3^2 is (-1)(3^2) = (-1)(9) = -9.
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 couldn't connect at all for the first week or so. I was ready to believe I couldn't 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 didn't know there was a rule for it. Since that discovery I've 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.