Mi ne komprenis la kaŭzon de la problemo, ĝis la instruistino montris la eraron sur la tabulo.

The ending -n marks the direct object in Esperanto: the thing directly affected by the verb.

• Mi ne komprenis la kaŭzon = I did not understand the cause
• la instruistino montris la eraron = the teacher showed the error

So:

• kaŭzon is what was not understood
• eraron is what was shown

But:

• de la problemo means of the problem, so problemo is inside a prepositional phrase, not a direct object
• sur la tabulo means on the board, so tabulo is also inside a prepositional phrase

A quick rule:

• direct object → often takes -n
• noun after a preposition like de, sur, ĝis clause, etc. → usually no -n

Here de means of and shows a relationship between two nouns.

• la kaŭzo de la problemo = the cause of the problem

This is the normal way to express possession or connection between nouns in Esperanto.

Why not da? Because da is used mainly with quantities, for example:

• glaso da akvo = a glass of water
• multe da laboro = a lot of work

But here we are not talking about an amount. We are talking about a relationship: one thing is the cause of another thing. So de is correct.

Ĝis means until.

In this sentence, it introduces the point in time when the situation changed:

• Mi ne komprenis ... ĝis ... montris ...
• I did not understand ... until ... showed ...

So the speaker’s lack of understanding continued up to the moment when the teacher showed the mistake.

It is very common to use ĝis this way before a clause:

• Mi atendis ĝis li venis.
• I waited until he came.

Because both actions are being described from a past-time viewpoint.

• komprenis = understood
• montris = showed

English also often uses past tense after until:

• I didn’t understand until the teacher showed the error.

Esperanto past tense is very straightforward:

• -is = past
• -as = present
• -os = future

So both verbs are in the past because both events happened in the past.

In Esperanto, ne usually comes directly before the word or phrase it negates.

Here it negates the verb:

• Mi ne komprenis = I did not understand

This is the most normal placement.

Compare:

• Mi ne vidis lin. = I did not see him.
• Ŝi ne venos. = She will not come.

If you move ne, you can change emphasis, but for learners the safest rule is:

• put ne right before the verb or the thing being negated

Instruistino is made from several meaningful parts:

• instru- = teach
• -ist- = a person professionally or habitually connected with something
• -in- = female
• -o = noun ending

So:

• instruisto = teacher
• instruistino = female teacher

This is very typical Esperanto word-building. Once you know the pieces, many words become easy to understand.

Examples:

• lernanto = student / learner
• kuracisto = doctor
• patrino = mother
• aktorino = actress

A modern note: some speakers may use instruisto for a teacher regardless of gender, and instruistino when they specifically want to say the teacher is female.

La is the Esperanto definite article, meaning the.

It appears before:

• la kaŭzon
• la problemo
• la instruistino
• la eraron
• la tabulo

It is used when the noun is definite, identifiable, or specific in the context.

So this sentence is talking about:

• a specific cause
• a specific problem
• a specific teacher
• a specific error
• a specific board

Unlike English, Esperanto has only one article:

• la = the

There is no separate word for a/an. If you want an indefinite sense, you usually just use the noun without an article.

Because sur means on, while en means in.

• sur la tabulo = on the board
• en la tabulo = in the board

When a teacher writes or points something on a blackboard or whiteboard, Esperanto normally uses sur la tabulo, just like English uses on the board.

Other common location prepositions:

• sur = on
• en = in
• sub = under
• apud = beside
• antaŭ = in front of
• malantaŭ = behind

Esperanto word order is fairly flexible, but not completely free. Because of endings like -n, the language can often show grammatical roles without relying as heavily on word order as English does.

The normal order here is:

• Mi ne komprenis la kaŭzon de la problemo, ĝis la instruistino montris la eraron sur la tabulo.

You could move some parts for emphasis, for example:

• Ĝis la instruistino montris la eraron sur la tabulo, mi ne komprenis la kaŭzon de la problemo.

That still works.

But beginners should usually stick to the common pattern:

• subject → negation → verb → object → other phrases

That sounds natural and is easiest to understand.

Both verbs end in -is, which marks the past tense in Esperanto.

• kompreni = to understand

• komprenis = understood

• montri = to show
• montris = showed

Esperanto verb endings never change for person or number. That means:

• mi komprenis = I understood
• vi komprenis = you understood
• li komprenis = he understood
• ili komprenis = they understood

The ending stays -is for all of them.

This is one of the simplest parts of Esperanto grammar.

Because Esperanto does not work like languages that require lots of special mood changes after certain conjunctions.

After ĝis, you normally just use the tense that matches the meaning:

• past if the action was past
• present if it is present
• future if it is future

Examples:

• Mi atendis ĝis ŝi venis. = I waited until she came.
• Mi atendos ĝis ŝi venos. = I will wait until she comes.

So in your sentence, montris is simply past tense because the teacher’s showing happened in the past.

A few letters may stand out to an English speaker:

• ŭ in kaŭzon is a very short w-like sound, usually part of a diphthong
  
  • aŭ sounds roughly like the vowel in English cow
• ĝ in ĝis sounds like the j in judge
• ŝ is not in this sentence, but often appears elsewhere and sounds like English sh
• ĉ sounds like English ch
• ĵ sounds like the s in measure

So approximately:

• kaŭzon ≈ KOW-zon
• ĝis ≈ jees with the initial sound of English jam
• instruistino has stress on the second-to-last syllable: in-stru-is-TI-no
• tabulo has stress on BU: ta-BU-lo

A useful general rule: Esperanto stress almost always falls on the second-to-last syllable.

The comma is natural and helpful here because ĝis la instruistino montris la eraron sur la tabulo is a subordinate clause, and the comma clearly separates it from the main clause.

Main clause:

• Mi ne komprenis la kaŭzon de la problemo

Subordinate clause:

• ĝis la instruistino montris la eraron sur la tabulo

In Esperanto punctuation, commas are often used to mark clause boundaries more consistently than in English. So the comma here is a good standard choice.

Yes, and that is exactly how the structure works in Esperanto.

Esperanto usually expresses this kind of relationship with de:

• la kaŭzo de la problemo
• literally: the cause of the problem

English can sometimes rephrase similar ideas with possessives, but Esperanto strongly prefers the de structure in cases like this.

For example:

• la pordo de la domo = the door of the house
• la komenco de la leciono = the beginning of the lesson

So this sentence uses a very normal and important Esperanto pattern.