> I find the Facebook comments interesting - lot's of "I can't figure this out". I suspect, strongly, that a lot of people learned the traditional manipulation

> I find the Facebook comments interesting - lot's of "I can't figure this out". I suspect, strongly, that a lot of people learned the traditional manipulation but never understood why they were doing it. If you really want students to understand why they are doing it, I see no other way than teaching methods for proving the correctness of algorithms, i. e. Hoare logic: https://en.wikipedia.org/wiki/Hoare_logic https://en.wikipedia.org/wiki/Hoare_logic