Every year when we study the graphing of linear inequalities in two variables, I get frustrated with my student's lack of understanding of inequality symbols. Of all the misconceptions that students learn in elementary school math, the alligator is the worst. You know what I'm talkin' about right??? The students learn that the alligator always eats the bigger number in order to help them understand the concept of an inequality.
Unfortunately, many students are completely stumped when determining if an ordered pair is a solution to an inequality when they end up with a statement like 0 < -6. They think that since the inequality is opening up toward the -6 that -6 is the bigger number and they shade their graph in the wrong part. It takes a lot of work to undo the concept of the alligator. I usually start with the number line and we talk about solutions of inequalities in one variable. The students have learned the "steps" in middle school, but they have no concept of what they are doing. They learn tricks for knowing which way to shade their number line, like "always draw your arrow the direction the symbol is pointing". I understand that teachers are trying to use terms that the student will understand, but I think if you give them enough visual examples and not just the "steps" they will eventually understand the concept of less than, greater than, and equal to.
Sorry for the rant today, my Algebra class really are going great and I don't have a lot to complain about, but gosh I hate that alligator and I just needed to tell someone.