Magistra dicit praemium bonae discipulae dari debere.

The sentence has two parts:

• Magistra dicit = The teacher says
• praemium bonae discipulae dari debere = the thing being said

That second part is an indirect statement in Latin. Instead of using a word like that, Latin often uses an accusative + infinitive construction.

So the structure is roughly:

• Magistra dicit
• praemium ... dari debere = that the reward ought to be given ...

And bonae discipulae tells you to whom it should be given.

Because Latin usually does not need one here.

After verbs of saying, thinking, knowing, hearing, and so on, Latin often uses indirect statement:

• verb of saying/thinking
• then a subject in the accusative
• then an infinitive

So instead of saying something like The teacher says that..., Latin says more literally:

• The teacher says [the reward to be given ought]

That sounds strange in English, but it is normal Latin.

Because they are doing two different jobs:

• dari = to be given
• debere = to ought / to have to, usually translated more naturally as ought to or should

Together, dari debere means:

• to ought to be given
• more natural English: ought to be given / should be given

So debere is the main infinitive of the indirect statement, and dari depends on it.

Dari is the present passive infinitive of dare.

• dare = to give
• dari = to be given

The sentence uses the passive because the reward is the thing receiving the action:

• the reward is given

not

• the reward gives

So praemium ... dari means the reward to be given or, more naturally, that the reward is given / be given, depending on context.

Debere adds the idea of obligation or appropriateness.

It can mean:

• to owe
• to be obliged
• ought to
• should
• sometimes must

In this sentence, the natural sense is:

• ought to be given
• should be given

So the teacher is not merely saying that the reward is given, but that it ought to be given.

Because the sentence is in indirect statement after dicit.

If this were a direct statement, Latin could say:

• Praemium bonae discipulae dari debet
  = The reward ought to be given to the good student

But after dicit, Latin changes the reported statement into an infinitive construction, so debet becomes debere:

• Magistra dicit praemium bonae discipulae dari debere

So debere is there because the whole reported clause is being put into indirect statement.

In meaning, praemium is the subject of dari:

• the reward is what is to be given

But in an indirect statement, the subject normally goes into the accusative.

So logically it is the subject of the infinitive, but grammatically it is in the accusative construction used after dicit.

A small complication: praemium is a neuter noun, and in the singular its nominative and accusative look the same. So the form itself does not show the difference. You recognize its role from the construction.

Here it is best understood as dative singular:

• bonae discipulae = to/for the good student

That makes sense because dari often takes a dative of the recipient:

• something is given to someone

Also:

• bonae agrees with discipulae
• both are feminine singular here

So bonae discipulae means to the good female student.

Formally, yes — the ending could also be genitive singular.

So in isolation, bonae discipulae could mean either:

• to the good student (dative)
• of the good student (genitive)

But in this sentence, the dative is much more natural, because dari strongly suggests a recipient:

• a reward ought to be given to the good student

If you took it as genitive, the sense would be something like:

• the good student's reward ought to be given

That is possible in theory, but much less likely unless the context clearly points that way.

Because the sentence means to be given, not to give.

Compare:

• dare = to give
• dari = to be given

Since the reward is the thing being handed over, Latin uses the passive infinitive:

• praemium ... dari = the reward to be given

If Latin used dare, the reward would wrongly seem to be the one doing the giving.

Yes. Latin word order is much freer than English word order.

This sentence could be rearranged in various ways without changing the basic meaning, for example:

• Magistra dicit bonae discipulae praemium dari debere
• Praemium bonae discipulae magistra dicit dari debere

The endings, not just the position, tell you what each word is doing.

The given order is perfectly normal. It puts praemium early, which gives it some prominence, and leaves debere at the end, where Latin often places an important verb.

The infinitives here are present infinitives, which usually show action or obligation at the same time as the main verb.

So:

• Magistra dicit ... dari debere

means roughly:

• The teacher says that the reward ought to be given
• in other words, she says this obligation exists now

It does not by itself mean that the giving has already happened. It expresses a present or ongoing obligation relative to dicit.