Big ideas and big ways of thinking in statistics? I'm moving to a new university for the fall semester, and I'll be teaching a statistics class for the first time. I'm familiar enough with doing statistics (my dissertation in math ed was a mixed-methods study with some quantitative components). However, I've never taught it before, so I don't yet have that level of knowledge about key statistics concepts and/or the "right" ways to think about them.
To give you a for-example: When I teach calculus, I know that I want people to leave understanding (among other things) that the subject is really mostly about creating and refining approximations (the limit process; linear approximation; the Riemann sum; etc. etc.).
So, in an effort to bootstrap my understanding a little bit, I thought I'd ask here: What are the big ways of thinking that are really useful for understanding and properly using statistics? 
If it's helpful to know, our book is OpenIntro's Intro Stats with Randomization and Simulation.
Thanks in advance!
(FYI, I've crossposted this to /r/math.)
 A: If a student walks away understanding the distinction between an estimand, an estimator and an estimate, along with the idea of a sampling distribution, you will have done an above average job. Equipped with these concepts, it should be clear that one must evaluate estimators based on how well they perform on average, in some relevant sense (such as coverage or mean squared estimation error). 
On a more personal note, I favor de-emphasizing testing and especially cook book recipes for statistical tests --- it encourages missing the forest for the trees and arguably is not especially relevant in many applications. 
A: The American Statistical Association has resources for statistics education, available here:
http://www.amstat.org/asa/education/home.aspx
...including, in particular, resources for undergraduate education.
Perhaps the "Guidelines for Assessment and Instruction in Statistics Education (GAISE) College Report" (available under the "Guidelines and Reports" heading) might be helpful. Oh, here is a direct link, why not:
http://www.amstat.org/asa/files/pdfs/GAISE/GaiseCollege_Full.pdf
This report seems relevant for your question because it lists 9 "big" goals for students. From there you can follow up to get more detail about those goals; recommendations for how to help students work toward the goals; pointers to further resources; etc. Good luck!
