Puella discit quomodo vaccam mulgere debeat, et mater eam laudat.

Because vaccam is the direct object of mulgere.

The action is to milk the cow, so the thing being milked is in the accusative case:

• vacca = a cow, nominative
• vaccam = a cow, accusative

Even though mulgere is an infinitive, it can still take its own direct object.

Mulgere means to milk and is the present active infinitive.

Here it depends on debeat:

• debeat = she ought / should
• mulgere debeat = she ought to milk

So the clause means that the girl is learning how she should milk the cow. Latin often uses an infinitive after verbs like debeo in the same way English uses to + verb.

Because quomodo vaccam mulgere debeat is an indirect question.

After verbs like discit meaning learns, Latin often introduces an indirect question with a question word such as:

• quomodo = how
• quid = what
• cur = why
• ubi = where

In Latin, indirect questions normally use the subjunctive, so debeat is subjunctive rather than indicative debet.

Quomodo means how.

It introduces the indirect question:

• quomodo vaccam mulgere debeat = how she should milk the cow

So discit quomodo... means she learns how...

Because the sentence is not simply saying how she milks the cow. It is saying how she ought to milk the cow or how she is supposed to milk the cow.

That idea of obligation or proper method comes from debeo:

• debeo = I ought, I should, I must

So:

• mulgat would mean she milks
• mulgere debeat means she should milk / ought to milk

This can suggest correct procedure, instruction, or duty.

Eam means her and it refers to puella, the girl.

In the second clause:

• mater = the mother, subject
• eam = her, direct object
• laudat = praises

So mater eam laudat means the mother praises her.

Eam is accusative feminine singular, which matches puella.

Because se is reflexive and refers back to the subject of its own clause.

In mater eam laudat, the subject of the clause is mater. If Latin used se, it would mean:

• mater se laudat = the mother praises herself

But that is not the meaning here. The mother is praising the girl, not herself, so Latin uses eam.

Because mater belongs to a different declension.

Latin nouns do not all follow the same pattern:

• puella is first declension
• vacca is first declension
• mater is third declension

So mater is simply the nominative singular form of the noun meaning mother. Its endings are different because it belongs to a different noun family.

Latin word order is much more flexible than English word order.

A very common tendency in Latin is to place the verb at the end, especially in subordinate clauses. So:

• quomodo vaccam mulgere debeat

is a very natural Latin order.

The sentence could not be translated word-for-word into normal English order. Latin relies more on case endings and verb forms than on position in the sentence.

Here are the main forms a learner would usually want to know:

• puella = girl
• disco, discere, didici = learn
• quomodo = how
• vacca = cow
• mulgeo, mulgere, mulsi, mulsum = milk
• debeo, debere, debui, debitum = ought, should, owe
• mater, matris = mother
• is, ea, id = he, she, it; that
• laudo, laudare, laudavi, laudatum = praise

Knowing these forms helps explain why the words look the way they do in the sentence.

It can be either, depending on context.

Latin present tense often covers both:

• learns
• is learning

So puella discit could mean:

• the girl learns
• the girl is learning

English chooses between simple present and progressive more often than Latin does.

It has two main parts joined by et:

1. Puella discit quomodo vaccam mulgere debeat

   • Puella = subject
   • discit = main verb
   • quomodo vaccam mulgere debeat = indirect question acting as the object of discit

2. et mater eam laudat

   • et = and
   • mater = subject
   • eam = object
   • laudat = verb

So the structure is:

• main clause
• subordinate indirect question
• coordinated main clause

That is a very common Latin sentence pattern.