Magistra existimat unumquemque modum utilem esse, quoniam quisque discipulus alio modo discit.

After a verb of thinking, saying, knowing, perceiving, and similar ideas, Latin often does not use a separate word for that. Instead, it uses an indirect statement construction:

• existimat = she thinks
• unumquemque modum utilem esse = that each method is useful

So Latin says, more literally:

• The teacher thinks each method to be useful

This is a very common pattern in Latin, and English speakers usually need time to get used to it.

Because it belongs to that indirect statement construction.

After existimat, Latin does not use a normal finite verb like est. It uses the infinitive instead:

• direct statement: Unusquisque modus utilis est = Each method is useful
• indirect statement after existimat: Magistra existimat unumquemque modum utilem esse = The teacher thinks that each method is useful

So esse is there because Latin is turning the statement into reported thought.

In an indirect statement, the subject of the infinitive goes into the accusative.

Here, the thing that is useful is unumquemque modum. Logically, that is the subject of esse, but because Latin is using indirect statement, it becomes accusative:

• normal subject: unusquisque modus
• in indirect statement: unumquemque modum

So this is not the direct object of esse. It is the accusative subject of the infinitive.

Because utilem agrees with modum.

Since modum is masculine singular accusative, the adjective describing it must match:

• modum = masculine singular accusative
• utilem = masculine singular accusative

So utilem is a predicate adjective in the indirect statement: it tells us what modum is considered to be.

Unumquemque is the masculine accusative singular form of unusquisque, which means each, each one, or every single.

Because it agrees with modum, it has to be:

• masculine, because modus is masculine
• singular, because it refers to each method one by one
• accusative, because it is the subject of the infinitive in indirect statement

So:

• dictionary form: unusquisque
• form here: unumquemque

Yes, both can mean each, but there is a slight difference in feel.

• unusquisque often has a more emphatic sense: each individual, every single one
• quisque is a common simpler word for each

So in this sentence:

• unumquemque modum = each individual method
• quisque discipulus = each student

The difference is subtle here. It is mostly a matter of style and emphasis, not a major difference in meaning.

No.

Magistra is the subject of existimat:

• Magistra existimat = The teacher thinks

The subject of esse is unumquemque modum, but because it is in an indirect statement, it appears in the accusative.

So the structure is:

• magistra = the one doing the thinking
• unumquemque modum = the thing being thought to be useful

That distinction is very important in Latin indirect statement.

Quoniam means because or since.

It introduces the reason for the teacher’s opinion:

• Magistra existimat unumquemque modum utilem esse = The teacher thinks every method is useful
• quoniam quisque discipulus alio modo discit = because each student learns in a different way

So the second clause explains why the teacher thinks that.

Because discit is in a regular subordinate clause introduced by quoniam.

Only the clause dependent on existimat is turned into an indirect statement:

• unumquemque modum utilem esse

But after quoniam, Latin uses a normal verb:

• quisque discipulus alio modo discit

So discit is a standard present indicative form meaning learns.

Both are present indicative active, third person singular.

• existimat = she thinks
• discit = he/she learns

In this sentence, the present tense is being used in a general or habitual sense:

• the teacher generally holds this opinion
• each student generally learns differently

So this is not just about one moment in time; it expresses a general truth or regular situation.

Alio modo is an ablative phrase meaning in another way or in a different way.

Word by word:

• alio = another / different in the ablative singular
• modo = ablative singular of modus, meaning way, manner, method

Latin often uses the ablative in expressions of manner or means, sometimes without a preposition. So:

• alio modo discit = he/she learns in another way

This is a very natural Latin expression.

Yes. They are both forms of modus.

• modum = accusative singular
• modo = ablative singular

In this sentence they do different jobs:

• unumquemque modum = each method
  Here modum is part of the indirect statement.
• alio modo = in another way
  Here modo is in an ablative phrase of manner.

So they come from the same noun, but the different cases give them different functions.

Latin word order is more flexible than English word order, because the endings show the grammatical relationships.

This sentence is arranged quite naturally:

• Magistra comes first: the teacher is the topic
• existimat comes early: we quickly learn what she is doing
• unumquemque modum utilem esse follows as the content of her thought
• quoniam then introduces the reason
• quisque discipulus alio modo discit explains that reason

Also, putting esse at the end of the indirect statement is very common in Latin.

So while the word order is not random, it is less rigid than English and often helps with emphasis and flow rather than basic grammar.