Teach 180: I Want Proof (Day 121)

A few days ago I was talking to a colleague about the fact that her Honors Geometry students weren't as curious this year.  When she asked the students if they wanted to know why something was true or why it made sense, most of them said "No".  These same students want their math teachers to tell them what strategy they should use or what the assignment is so they can do it and get it over with.  These students say, "But how are we supposed to know how to do that problem on the test. We didn't have any like it on the homework."

When I was in high school, I always wanted to know why something was true and my teachers were very good out outlining steps to a proof or leading us in reasoning why something was true.  Just because some of my students don't share the same curiosity I did doesn't mean that I shouldn't expose them to mathematical arguments.  In fact, one of the Common Core State Standard for Mathematical Practices states that "Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments." In addition, "Students at all grades can listen to or read arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments." 

Most of my Calculus students are seniors and simply want to be told formulas for the derivative.  Few of them care to understand why the derivatives of f(x) = a^x and f(x) = log_ax are what they are.  The fact that they don't care, doesn't mean that I should not share the argument with them.  In class today, I presented the following argument to show that the derivative of f(x) = a^x is (ln a)(a^x).  Enjoy!