Rapke (2008) explores the reasons why $(-1)(-1)=1$ and what knowledge must be known to explain it. I liked how she divided her findings into two sections: Pseduoreasoning,
and formal explanations. In order to rigorously prove $(-1)(-1)=1$, background information on the distributive property and other axioms
(but not really) of the real numbers is required.
The point Rapke makes is that when teaching children mathematical axioms, they must be taught why. Teaching a student the reason why a property holds will help them build skills as recognizing patterns. If we say, believe for now
then the student will just accept the fact without question, and will only regurgitate that information on a test.
There were many different approaches to the explanation of why $(-1)(-1)=1$, including word problems, but Rapke says that these word problems are inaccurate and may be falsely logical. It cannot help the student identify a similar situation and use the same concept. Another method is that students may use patterns to predict solutions. Although this is not a formal proof, it can provide for some great intuitive reasoning and even lead to inductive reasoning. I never realized, however, that it is not ideal because because sometimes the pattern may not work. For example, when trying to find $0^0$, the student may write this:
$$ (-2)^0=1\\ (-1)^0=1\\ 0^0=?\\ 1^0=1\\ 2^0=1\\ \therefore 0^0=1 $$
In other words, $\lim_{x\to 0} x^{0} = 1$. Following this pattern, students were led to believe that $0^{0}=1$. Since $0^0 \neq 1$, the patten fails. Sometimes using patterns does not lead to correct results, regardless of the intuitive benefits they provide.
In addition, students may develop a pattern for the wrong reason when trying to solve a problem. Using patterns to explain concepts is not ideal because teachers must explain which patterns are okay to use, which are not, and which have exceptions.
Rapke says that the most ideal method of proving the $(-1)(-1)=1$ problem is to use the distributive property. Furthermore, all algebraic properties should be used, because all algebraic properties hold across the number systems real numbers, which is the number system used in 99% of high school mathematics.
References
- Rapke, T. (). Thoughts on why (-1)(-1)=+1. Mathematics Teacher, 102(5), 374–376.