CPI And Inflation Are We Teaching It Right?

The following is a question from a student back in March 2006: "If the CPI overstate inflation, how do we know we are not overcorrecting for this and end up understating inflation? What is the number telling us about the real world when we remove the volatile part of inflation? Does that mean nobody uses energy during that time!?"
I really liked this question because it stumped me. I started thinking about this question because a Math teacher asked me about COLA.
Should we ask students to evaluate various types of Price Indexes? And chose the one that will help them with getting a raise? Or if they move into management select the number that would justify denying their workers a raise!