In the verb “to teach”, is the student who is being taught the indirect object and the subject matter the direct object? I would think so: A sentence like “The professor taught the students English” could be rewritten as “The professor taught English to the students”, so “the students” would appear to be the indirect object and “English” the direct one. What makes me doubt, though, is that a sentence such as “I’ll teach you” or the imperative “Teach me” appears to be grammatical, yet it has only one object - which conflicts with the rule that you can have a direct object without an indirect one. Are these simply exceptions from the rule that an indirect object also needs a direct one, or is the theory that the direct object is there, only elided?
ISTM this word, like many, has distinct (but related) meanings:
- To pass on knowledge about something
- To pass on knowledge to someone
I’m quite sure that, in certain other languages, these are expressed with entirely distinct words (rather how English “is” is expressed by Spanish “estar” or “ser,” depending on the nuance of meaning not obvious to English speakers.)
When either “teach” is used by itself, it takes a direct object (the subject matter or the “pupil,” respectively).
When the word does double duty in a single sentence, English forces the “pupil” into the indirect object slot, keeping the subject matter in the direct object slot. Interesting! (Note how the sentence could be recast as “He teaches math to the girl.”)
(Many languages make a stronger distinction between direct objects of verbs that actually affect them – like throwing a ball or burning down a house – and of those that just passively have something done to them, like teaching. The ergative languages like Yucatec Maya treat this as a fundamental distinction, while English tends not to. Languages like Spanish are in between – they often use the reflexive to express the less active sort of verb-object relationship).
I thnk you’re right, though I am no grammar expert. I think it’s similar to a verb like “to write,” as in “I wrote my grandmother” or “I wrote my Congressman” (with the implied direct object being something like “a letter”).
But it’s not as important a distinction in English as in German, where there are different articles and pronouns for the dative vs accusative case.
Yeah, “to teach” is a somewhat unusual verb in that it has both transitive and intransitive usage, leading to possible ambiguity.
“I teach math” = math is the subject I’m teaching
“I teach Karl” = I am teaching (something to) Karl
“I teach Karl Marx” = either I am the teacher for a class on the subject of Karl Marx, or I am a teacher and I have a student named Karl Marx
“What do you do for a living?” / “I teach” = intransitive, no object needed, but the meaning is clear
English is weird! 
Never mind.
Indeed. A similar (though not identical) situation is with “marry.” If a minister named Betty is married to a man named Fred, and she officiates the wedding of Greg and Susan, we can either say “Betty married Fred,” or “Betty married Greg and Susan.”
In this case, though, we can’t say “Betty married Greg, Susan, and Fred,” except in jest. We can’t let the word do double duty.
But what if Betty officiated her own wedding? (Is that allowed by any church or legal system?) Can we say “Betty married Greg Greg”…in which, like with “teach,” the first Greg is now an indirect object? Of course we can’t! Not the same after all.
Brings to mind the difference between “England fought with France” and “England fought with France against Germany.”
Yes! Good example. Surely some linguist has coined a term for a word whose varied meanings each takes a different sort of object.
There’s probably also a term covering the (now somewhat old fashioned, but in currency when I was in junior high) use of “teach” as a familiar shortening of “teacher,” as in, “hey teach, what page are we on again?”
It’s sort of akin to synecdoche, except instead of referring to a partial element of the physical whole (e.g. “twenty sails” = a fleet of twenty ships), the teacher is reduced to the action performed.
Or is that a matter of referring to the person by their job title?
“Hey driver, when are we going to get there?” Is “driver” their function or their job title? If I change it to “Hey Driver, when are we going to get there?” how does that alter matters? And how would anyone know which casing you meant when speaking, not writing?
FYI in US English “Driver” is the term of art / mutual respect that professional truckers use to refer to one another. Given that their life on the road involves lots of brief one-time encounters between passing strangers at truck stops and such, nobody bothers much with introductions.
“The professor taught the students English.” Yes. “The Students” is the indirect object.
[Ack. Removed non-factual comment.]
English is weird.
Both of these are purely American English.
I would say, “I wrote to my grandmother” etc,
English is full of what would be understood words that we don’t say in order to shorten our speech.
“I gave him the news” is essentially “I gave the news to him”. Indirect object is just a cute way of saying “leave out the preposition”. Imperative uses the implied “you” as in “get out!” which is actually “You get out!”.
After all, what about “I talked to him”? “I told him”? “I told him the news”?
English is weird.
I am really not convinced that there is any value in distinguishing direct and indirect objects in English. Teach can be described as having one or two complements, one of which might be a prepositional phrase. Semantics is where all the distinctions are. Some verbs take a whole sentence as a complement. Consider “I enjoy taking the kids to the movies”. There is even a verb that takes three complements. Bet you a buck you can’t guess what it is.
Once upon a time, Old English had inflected nouns with nominative, genitive, dative, and accusative forms, singular and plural, and direct and indirect object made sense. But the inflections are gone and I don’t think it helps to impose that grammar on modern English. It was imposed by people who thought Latin was the perfect language. I ask, if that was why did the daughter languages mostly lose those inflections?
I think that’s a good point. As you know, this attempt to force English to fit Latin grammatical expectations (e.g., noun case endings) has a long and durable history. Another example is “it’s me” – perfectly good English (and French), but appears to violate the Latin case ending system (if we pigeonhole “me” as akin to a Latin accusative). Some influential, Latin-loving curmudgeon a couple centuries ago insisted it’s bad English, and so even today kids are told to say “it is I” - though nobody actually does.
(Interesting how “it’s I” sounds even worse. Maybe because the stilted, formal feel of using “I” here only goes with the stilted, formal feel of avoiding contractions.)
I can’t stand the suspense! (Now there’s an interesting verb-complement relationship. I’m not doing anything to the suspense…)
Wait! ISWYDT! The verb is “bet”! 
You got it. Aside from a synonym like wager, I can’t think of another example. And the third complement is a sentence.
If Latin was actually any damned good, we’d still be speaking it. We don’t, so it isn’t. Harrumph! 
We are still speaking Latin – maybe not you and I, but people are. Italian, Spanish, etc, are probably as close to Latin as modern English is to Old English
But the Latin they’re close to is not the Latin taught in school. That’s classical Latin, which was the variant cultivated by scholars of the first century BC and AD. The Romance languages of today derived from post-classical Vulgar Latin, which already exhibits significant differences to classical Latin in grammar and vocabulary.
QED!