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

Because uterque, utraque, utrumque is a special word meaning each of the two or, in smoother English, both.

It is singular in form, but its meaning already includes two. So Latin can say:

• uterque modus = each of the two methods / both methods
• utrumque modum = the accusative form of that phrase

So even though English normally uses a plural (both methods), Latin often uses uterque with a singular noun.

In this sentence, utrumque modum is:

• masculine
• accusative
• singular

because it matches modum, from modus.

This happens because existimat often introduces an indirect statement.

In English we say:

• The teacher thinks that both methods are useful.

In Latin, after verbs like existimat (thinks, judges, believes), Latin usually does not use a separate word for that. Instead it uses:

• accusative subject + infinitive

So:

• utrumque modum = the subject of the indirect statement, put in the accusative
• esse = to be
• utilem = predicate adjective agreeing with utrumque modum

So literally the Latin is something like:

• The teacher thinks both-method to be useful

That is normal Latin, and idiomatic English turns it into:

• The teacher thinks both methods are useful.

Because in an indirect statement, the adjective that describes the accusative subject also goes into the accusative.

Here the accusative subject is utrumque modum, so the adjective describing it must match:

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

If the sentence were direct instead, you would get nominatives:

• uterque modus utilis est = each/both method(s) is/are useful

But after existimat, the structure changes to indirect statement:

• utrumque modum utilem esse

Because classical Latin normally does not use a conjunction like English that in this kind of sentence.

English:

• She thinks that...

Latin:

• She thinks [accusative + infinitive]

So instead of a separate word meaning that, Latin signals the idea through grammar:

• existimat
• then utrumque modum utilem esse

That whole accusative-infinitive phrase is the content of what she thinks.

Quoniam means since or because.

It introduces the reason for the teacher’s opinion:

• Magistra existimat utrumque modum utilem esse
• quoniam quisque discipulus alio modo discit

So the sense is:

• The teacher thinks both methods are useful, because/since each student learns in a different way.

A learner may also know quia for because. In many contexts, quoniam and quia can be quite close in meaning.

Because quisque means each or each one.

Latin often expresses a general idea distributively, one individual at a time:

• quisque discipulus = each student
• literally: every student considered individually

English often prefers:

• students learn in different ways

But Latin is perfectly natural with the singular:

• each student learns in a different way

This wording emphasizes individual differences.

Alio modo is ablative singular:

• alio = ablative singular of alius (other, another)
• modo = ablative singular of modus (way, method, manner)

Together it means:

• in another way
• by a different method
• differently

This is a very common Latin pattern: an ablative phrase used adverbially to describe how something happens.

Similar expressions are:

• hoc modo = in this way
• eodem modo = in the same way
• nullo modo = in no way

Yes. They are both forms of the same noun, modus.

In this sentence:

• utrumque modum: modum is accusative singular
• alio modo: modo is ablative singular

They have different endings because they do different jobs:

• modum is part of the indirect statement after existimat
• modo is in an ablative phrase meaning in a different way

So Latin is using the same basic noun twice, but in two different grammatical roles.

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

This sentence is arranged quite naturally:

• Magistra = topic/subject first
• existimat = main verb
• utrumque modum utilem esse = what she thinks
• quoniam... = reason clause

The order also gives a nice flow:

1. who is doing the thinking
2. what is thought
3. why

You should not expect Latin always to match English order. The endings matter more than position.

Yes. In fact, that is often the most literal explanation.

• uterque = each of the two
• natural English in many contexts = both

So here:

• utrumque modum utilem esse

can be understood very literally as:

• that each of the two methods is useful

But smoother English will usually say:

• that both methods are useful

Both translations capture the idea. The more literal one helps you understand the grammar; the smoother one sounds more natural in English.