Mathematical Fallacy Analysis I am interested in mathematical fallacy and found some cases about it. I am one of education major college student, and of course I am afraid that students in school will encounter it, especially the lack of understanding ones. Here is the sample. 
We already know that $(-1)^3 = -1$. Yet, I will show you that it is not a true fact. 


  $(-1)^3 = (-1)^{\frac{6}{2}} = ((-1)^6)^{\frac{1}{2}} = 1^{\frac{1}{2}} = 1$


So, the conclusion is $-1 = 1$. 
Most of students are easily trapped by such a imaginary number cases and absolute value properties, and that mathematical fallacies I shall look at are in that areas. 
What I want to know is that: are there any method, study theory, approachment, or anything, which can be used by teacher to make students have capabilities to analyze mathematical fallacy in solution steps and it is more better if they can think critically, implied that they just don't memorize certain math subject's properties.
The example given (or others like it) is exactly what students should be encouraged to keep in mind when they work with fractional exponents, because it demonstrates how easy it is to go astray if you don't pay attention to the precise statement of mathematical laws.

The fallacy, of course, lies in the naive expectation that the "associative" law $z^{ab}=(z^a)^b$ holds for all numbers $z$, not just the positive reals.  (Note, I'm assuming here that $a$ and $b$ are real numbers; the situation is even more delicate if $a$ and $b$ are complex.) It could be worth having students concoct their own examples of the fallacy, so that they can take "ownership" of it. For students familiar with function notation, it might also be worth presenting it in the form of an abstract question:  Is $E(E(x,y),z))$ necessarily equal to $E(x,M(y,z))$? (I trust it's obvious what operations $E$ and $M$ refer to.)
The example given (or others like it) is exactly what students should be encouraged to keep in mind when they work with fractional exponents, because it demonstrates how easy it is to go astray if you don't pay attention to the precise statement of mathematical laws.

The fallacy, of course, lies in the naive expectation that the "associative" law $z^{ab}=(z^a)^b$ holds for all numbers $z$, not just the positive reals.  (Note, I'm assuming here that $a$ and $b$ are real numbers; the situation is even more delicate if $a$ and $b$ are complex.) It could be worth having students concoct their own examples of the fallacy, so that they can take "ownership" of it. For students familiar with function notation, it might also be worth presenting it in the form of an abstract question:  Is $E(E(x,y),z))$ necessarily equal to $E(x,M(y,z))$? (I trust it's obvious what operations $E$ and $M$ refer to.)