prepositions - "like I" or "like me"?

In high school we learned to say "than I" and "as I" because you could potentially add an "am" to the end of the sentence. Examples:

"She is smarter than I." (Think: "...than I am.")
"He is as tall as I." (Think "...as I am.")

So analogously, shouldn't it be "like I" as well:

"He is sincere, just like I." (Think: "... like I am.") But universally, it seems that we use "like me". Where does this reasoning break down? Is there history here?

Answer

The rules you were taught are artificial. It is very rare to hear "as I" used in the way you have it in your examples out in the wild. It may be correct according to prescriptive English grammar, but it is not idiomatic to the language until you add the extra bits. One would say either:

She is smarter than me.

or

She is smarter than I am.

The same pattern emerges when looking at like. Used by itself, in idiomatic English, you would use the object pronoun when the word is used alone, and the subject pronoun when the phrase extends into a sentence-like structure.

Despite the vain longings of those few people who want English to be nice and neat (and have a one-to-one correspondence with Latin and Greek), our language has its wrinkles and inconsistencies. This is one of them.