(ick, could we get a little more SEO-y with that title?)
Maybe you're like me and always heard of "new math" as some horrible misstep America made in the 1960s; or maybe you've bought into the ads against what people are saying about the current curriculum trying to meet the Common Core Standards.
I'd like to think that Common Core addresses many of the drawbacks of "New Math", but it definitely isn't doing much better in the zeitgeist than New Math did, and a HUGE part of that is having teachers that aren't trained in the mathematics they're expected to teach.
Having teachers that aren't able to balance educational heuristic techniques, definitions, and explanations with more rigorous ones is definitely an issue that needs to be addressed within each individual curriculum based on CCSS. You can't go the way we've all heard New Math went and just throw college-level definitions at students right off the bat (maybe if they're adults that are interested in their education, but even then the heuristic explanations should be included as well).
But you also can't do what teachers have seemingly always done: either completely ignore the actual definitions of mathematical objects and rely on incorrect definitions and arcane rules or include the definitions only as something to be memorized or as an aside with no relevance.
We have rigor in mathematics for a reason, and if we didn't need it or if it wasn't useful, we wouldn't have it. That's how math works. (That's one of the reasons why I will argue that 0^0=1 any time we need it to be and that a parallelogram is also a trapezoid).
Rigor is both needed *and* useful.
I've written more about this, but don't want to mess with the math formatting, so please follow the link below for a pdf illustrating my argument through the context of absolute value of a number. Don't forget to come back and comment/+1/share when you're done. (Note: if you are came here just to get help on learning how to solve absolute value equations and absolute value inequalities, the "teaching" portion of the essay starts on page 5, but the article is primarily about rigor and teaching strategies).
I hope this explains how you were probably taught absolute value and why you were taught that way, what we can do with the definitions if they are given to us, and the importance of teaching rigorous definitions in mathematics. If you have any questions, please comment below. Thank you!