Krítin brotnaði, en kennarinn hélt samt áfram að skrifa á töfluna.

The -in is the suffixed definite article, so krítin means the chalk.

• krít = chalk
• krítin = the chalk

In Icelandic, the definite article is usually attached to the end of the noun instead of being a separate word like English the.

Here, krítin is the subject of the sentence, in the nominative singular.

Because brotna and brjóta are different verbs.

• brotna = to break, to get broken (intransitive: the thing itself breaks)
• brjóta = to break something (transitive: someone breaks it)

So:

• Krítin brotnaði = The chalk broke
• Kennarinn braut krítina = The teacher broke the chalk

In this sentence, the chalk is not being shown as directly broken by someone; it simply broke, so brotnaði is the correct form.

Brotnaði is the past tense, 3rd person singular, of brotna.

So Krítin brotnaði means The chalk broke or The chalk got broken.

A quick comparison:

• Krítin brotnar = The chalk breaks / is breaking
• Krítin brotnaði = The chalk broke

Because en means but, and it joins two clauses:

• Krítin brotnaði
• kennarinn hélt samt áfram að skrifa á töfluna

Using a comma before en is normal in Icelandic when connecting two full clauses, just like in English you often write ..., but ....

For the same reason as krítin: it is the suffixed definite article.

• kennari = teacher
• kennarinn = the teacher

So both main nouns in the sentence are definite:

• krítin = the chalk
• kennarinn = the teacher

Hélt áfram means continued or more literally kept going on.

It comes from the verb phrase halda áfram = to continue.

Its parts are:

• hélt = past tense of halda
• áfram = on, onward, further

So:

• hann heldur áfram = he continues
• hann hélt áfram = he continued

In this sentence, kennarinn hélt áfram means the teacher continued.

They do different jobs.

• áfram is part of the verb phrase halda áfram = continue
• samt means still, nevertheless, or all the same

So the sentence is not just saying the teacher continued. It says the teacher continued despite that.

A natural sense is:

• The chalk broke, but the teacher still / nevertheless continued writing on the board.

Without samt, the sentence would simply say the teacher continued. With samt, it adds the idea of even so.

Because halda áfram is commonly followed by að + infinitive.

So:

• halda áfram að skrifa = continue to write
• halda áfram að tala = continue to speak
• halda áfram að vinna = continue to work

Here:

• skrifa = to write
• að skrifa = to write after this verb pattern

This is a very useful structure to learn: halda áfram að + infinitive

This is about the preposition á, which can take different cases.

• á + accusative often suggests movement toward a surface or onto something
• á + dative often suggests location on something

Here, töfluna is accusative singular of tafla (board), so á töfluna literally has the sense of writing onto the board.

Compare:

• skrifa á töfluna = write on / onto the board
• textinn er á töflunni = the text is on the board

English usually just says on the board in both cases, but Icelandic often shows this distinction through case.

Töfluna is the accusative singular definite form of tafla.

The forms are:

• tafla = a board (nominative singular)
• taflan = the board (nominative singular definite)
• töfluna = the board (accusative singular definite)

The vowel changes from a to ö because of Icelandic noun inflection.

This is normal Icelandic word order.

In the second clause:

• en = but
• kennarinn = subject
• hélt = finite verb
• samt = adverb
• áfram = part of the verb phrase
• að skrifa = infinitive phrase

So the structure is:

en + subject + finite verb + adverb + áfram + infinitive

Placing samt after the finite verb is very common. Icelandic often puts sentence adverbs like this in that position.

Yes, depending on context, samt can feel like still, anyway, all the same, or however / nevertheless.

In this sentence, the best natural English choices are probably:

• The chalk broke, but the teacher still continued writing on the board.
• The chalk broke, but the teacher nevertheless continued writing on the board.

So samt adds contrast: something went wrong, but the teacher carried on anyway.

It would become more like chalk and a teacher rather than the chalk and the teacher, but the exact forms would also need to change for case and sentence grammar.

Very roughly:

• Krít brotnaði = Chalk / a piece of chalk broke
• kennari hélt áfram... = a teacher continued...

In the original sentence, the definite forms make it sound like we are talking about specific things already known in the situation:

• the chalk
• the teacher
• the board

Yes, that is a good idea.

Even though you can analyze each part separately, skrifa á töfluna is a very common classroom expression meaning write on the board.

Useful related expressions:

• skrifa á blað = write on paper / on a sheet
• skrifa í bók = write in a book
• skrifa á vegginn = write on the wall

Learning whole chunks like halda áfram að skrifa á töfluna will help you sound more natural than translating word by word every time.