Did I mention that this is my first year and I'm straight out of college and don't really know anything about how to help people learn things? OK, just getting that out of the way.
There are better and worse ways to help people understand things. Here is some of what I've learned about that over the past few months.
1) Verbs are important. When you see 4 x's in the numerator and 2 x's in the denominator are you inclined to cancel, simplify or unmultiply the fraction? One of these verbs is noxious, the other annoying and one tells a student exactly what they should think about doing. This stuff matters. Man, if I have to tell another Algebra 2 student not to cross out a summand in the numerator even though I thought we cancel stuff when it's on the top and bottom of a fraction I swear to God that larynxes will be torn out of children's...happy place, happy place...OK, I'm in control. I'm just going to be careful when I talk to my freshmen, that's all. They're not going to hear the word "cancel" once. But they will hear me talk about unmultiplying fractions, and they'll know exactly what I want them to do.
2) Some procedures are better than others. So, how do you solve a rational inequality, or an absolute value inequality, or a quadratic inequality? You want to give them a procedure that will yield them the correct answer. But you also want them to understand why a procedure works. So you choose, as your procedure, to teach them to solve the inequality like an equation, plot those points on a number line, and then to test each region between those points to see if it satisfies the inequality. And, since you want them to understand all of this, you don't give them the procedure before you explain to them how it works. Right?
Blech. Students quickly forget your explanation, and just rely on the procedure, which is a brainless algorithm that doesn't put the mind in contact with the relevant concepts? But what else can you do?
Well, I've learned that I can design my own procedures, and that with subtle changes I can design them so that they force a student to come in contact with actual math. So instead of the above procedure for solving inequalities, I now teach students to split up the inequality into two functions, to graph both of them, solve the system of equations to find where the two curves intersect and then graphically intuit which regions satisfy the inequality. That's the same procedure as above, in case you're following, just rewritten from code into math for humans.
I suppose that, in sum, my basic moral from my first semester of teaching is: put students in contact with the right ideas early and often.