You do not have to be on Twitter or Facebook long before you come across some glaring example of pedagogical malfeasance.
Today some poor devil mentioned a teacher who cut one letter grade from a paper each time the word that occurred, a lesson the poor devil has apparently carried into adult life.* One recalls Jane Austen: "We all love to instruct, though we can teach only what is not worth knowing."
Let's be clear.
That as a relative pronoun introduces dependent clauses, marking the relationship with a main clause.
It is often omitted in speech, particularly when the main clause and dependent clause are both short, as in the line from O Brother, Where Art Thou?: “We thought you was a toad.” But you do not want to make this a universal practice.
The Associated Press Stylebook, so frequently cited by people unaware of what is in it, offers useful advice. First, that must be used when an adverb of time follows the main verb. "They found out Thursday that their ship had sailed."
Also, some verbs idiomatically require the use of that: advocate, assert, contend, declare, estimate, make clear, point out, propose, and state.
And it must be used before subordinate clauses beginning with after, although, because, before, in addition to, until, and while. "McIntyre said that while he has made this point repeatedly over the years, no one appears to have taken notice."
AP advises, “When in doubt, include that. Omission can hurt. Inclusion never does.”
For my part, it is depressing to calculate how much time I spent over the past forty years inserting that into awkwardly phrased and unidiomatic newspaper copy.
* I do not know but suspect that the class was in journalism, since journalists as a class appear to get the fantods over that relative pronoun.