How to demonstrate excellent undergraduate and graduate teaching capabilities in academic position applications? It is quite common in almost all position opening announcements in academic institutes to demonstrate EXCELLENT teaching capabilities. For instance sentences like below:


  Applicants are expected to demonstrate excellent undergraduate and graduate teaching capabilities 


But the question is HOW to demonstrate that excellence? Is just listing subjects taught during last couple of years sufficient? I guess we can use teaching statement or teaching philosophy to demonstrate that ability too. But, what type of sentences show that excellence in teaching statement?
It is a great question that I have asked myself in the last several years and I'm also looking for an informed answer. In the mean while, I have come up with an idea to demonstrate my teaching abilities to potential employers: 

After graduation, I taught several graduate courses mainly on mathematical analysis, such as measure theory, harmonic analysis, C*-algebra. Based on these experiences, I have written a lecture note on C*-algebras which is available in arXiv. I thought if a serious employer really wants to evaluate my teaching abilities, nothing speaks better than my lecture notes.