I've recently taken on the task of helping out in my school's Math Center. The courses I assist in range from Algebra to Calculus. While I'm younger (in my 20's), most of the students at the school are 40+.
The first students I worked with were taking a Calculus Prep class. Teaching them was fairly easy - most of what I did was show them that the steps required to solve a problem were steps they already knew how to do and were familiar with - the problem was just being presented in a unique manner.
However, I recently got tasked with helping out students taking the lowest level math we offered - and I found it surprisingly difficult. Teaching a younger child math, you can tell them that a rule is a rule, and that they must accept it in math and go forward. From there, they might start to test that rule, and see that it applies in all cases.
But, when teaching adults, I've found that I can't just tell them "this is the way it's done, get used to it." I have students that are struggling with topics like adding and subtracting negatives, or determining if a function is odd or even - essentially why $-(x^2)$ is different from $(-x)^2$. I can try to explain negatives using a number line, or walk them through PEMDAS again, but it just doesn't seem to stick.
It seems like children just need to know how, but adults need to know why.
Any advice for teaching simple concepts and rules to adults, specifically working with negatives or proper application of PEMDAS?