Depth of Knowledge

One of my real stumbling blocks this year throughout all of our school improvement, leadership team, instructional team, student support team, and graduate class meetings has been "What do I teach?" Anytime I look through textbooks or online for resources, I find different ways of presenting the same concept. But hey that's differentiation and multiple intelligence styles and so on and so forth, right? What is messing me up is wondering if my students could solve any problem thrown at them pertaining to a specific concept I've taught.

For example, I felt like a did a good job of teaching systems of equations. We did graphing on the calculator, we did word problems, the whole kit and kaboodle. As I was creating the test, I found a problem where one of the equations had a variable on both sides, some like 2x = 45-5y. We had done many examples but not one with variables on both sides. Oh, but we solve regular equations like that all the time, right? So I put it on there. I had a ton of questions on that one. Some said it was infinite and some said no solution and some canceled out the variables altogether. Not everyone got it wrong but fewer knew how to handle it than I would have liked.

I know I've posted before on increasing students' critical thinking skills and I guess I can attribute some of this to that issue but my real question is "How do I teach to mastery so that students can apply the concept anywhere?" This is probably an obvious question to most or even the definition of teaching to some. But for me, it's my latest realization. Of course I'm always thinking about test scores but truly, how do we ever know what depth to teach to? Our district has a chronic problem of repeating material over and over without ever going any deeper. We're currently trying to align things to Common Core as well as K-12. But no matter what list of standards we look at, I could probably say I teach all those things. The problem is, have I taught them to the same depth of understanding that the ACT is assessing? The obvious thing is to look at the ACT but even then, I'm not sure how much that can help. If a bunch of students get a question right, they throw it out. If a bunch of students get a question wrong, they throw it out. It's impossible to know exactly what will be on the test and what it will look like.

Does the answer lie in assessment, critical thinking, more practice, or better teaching? Do my students need harder problems and more time to think things through? How can I ever be sure that they can apply the concept in new situations that we have not practiced together? Is that an okay expectation to have? Isn't that the definition of math or of learning- using what you know to figure out something you didn't previously know?

I feel frustrated sometimes because even if we had the perfectly aligned harmonious pacing guide, how would we be sure that each teacher is teaching to mastery so that students can reapply the concept in new situations throughout their mathematical career?

For example, addition. Students learn addition and it carries them through many new situations. We throw in variables, exponents, square roots, absolute value, and even imaginary numbers. But for the most part, do students ever not know how to add, even in new situations? Is it due to the fact that they've been adding for years and years? Is it the fact that no matter the situation, adding itself never changes? How can that work for other concepts?

How can my students become so comfortable with a tool that they can figure out how to use it in any situation?