Vi er i gang med at læse om nutid og datid, mens læreren skriver flere eksempler på tavlen.

Er i gang med at + infinitive is a very common Danish way to say that an action is in progress.

So:

• Vi læser = we read / we are reading
• Vi er i gang med at læse = we are in the middle of reading / we are busy reading

Danish does not have a separate -ing tense like English does, so this expression is one useful way to make the ongoing nature of the action extra clear.

Because at marks the infinitive, like to in English.

• at læse = to read
• at skrive = to write

In the expression i gang med at + verb, the at is required:

• Vi er i gang med at læse
• Han er i gang med at skrive

So here, at læse is the infinitive after the fixed phrase i gang med.

Læse om means read about.

So:

• læse = read
• læse om = read about

This is just the normal combination in Danish when you are reading on a topic:

• læse om historie = read about history
• læse om nutid og datid = read about present and past tense

You should learn læse om as a common verb + preposition combination.

They are grammar terms:

• nutid = present tense
• datid = past tense / preterite

These are standard Danish school grammar words. In this sentence, they are being used as general concepts, so there is no article.

That is why you get:

• om nutid og datid

not necessarily:

• om nutiden og datiden

The version without the article sounds more like naming grammar categories in general.

Because they are being used as abstract grammar terms, not as specific countable nouns.

In English, you also often say things like:

• We are studying present and past tense

rather than always saying:

• the present tense and the past tense

Danish often does the same with subject labels and abstract categories.

Mens means while.

It connects two actions that happen at the same time:

• Vi er i gang med at læse om nutid og datid
• mens læreren skriver flere eksempler på tavlen

So the idea is:

• We are reading about present and past tense while the teacher is writing more examples on the board.

It introduces a subordinate clause.

Because mens introduces a subordinate clause, and subordinate clauses in Danish do not use the normal main-clause inversion pattern.

Main clause:

• Læreren skriver flere eksempler på tavlen.

Subordinate clause after mens:

• mens læreren skriver flere eksempler på tavlen

So the subject læreren comes before the verb skriver.

A useful thing to remember:

• In Danish main clauses, the finite verb is typically in second position.
• In subordinate clauses, the word order is usually more straightforward, with the subject before the verb.

Because læreren means the teacher.

Danish usually adds the definite article to the end of the noun:

• en lærer = a teacher
• læreren = the teacher

So here it refers to a specific teacher already understood from the situation.

Flere means more or several more, depending on context.

• et eksempel = an example
• eksempler = examples
• flere eksempler = more examples

In this sentence, it means the teacher is writing additional examples.

Note that flere is the plural/comparative form used with countable nouns:

• flere bøger = more books
• flere elever = more students

På means on, and tavlen means the board.

So:

• en tavle = a board
• tavlen = the board
• på tavlen = on the board

This is the natural Danish expression for writing on a classroom board, whether it is a blackboard or whiteboard.

In practice, you will often see a comma before mens, as in this sentence.

Danish comma rules can vary depending on comma style, but many learners will meet sentences written with a comma before subordinate clauses like this one. So the comma here is very normal and not surprising.

The important grammar point is that mens læreren skriver flere eksempler på tavlen is a clause attached to the first part of the sentence.

Yes. That would also be correct.

• Vi læser om nutid og datid = We read / we are reading about present and past tense.
• Vi er i gang med at læse om nutid og datid = We are in the middle of reading about present and past tense.

The longer version puts more emphasis on the action being ongoing right now. Danish often leaves this unstated unless it matters, so both versions are possible depending on nuance.

Yes:

• Mens læreren skriver flere eksempler på tavlen, er vi i gang med at læse om nutid og datid.

That is also correct.

When a subordinate clause comes first, the main clause still follows normal Danish main-clause word order, so you get:

• ..., er vi ...

with the verb er before the subject vi in the main clause. This is a very important Danish word-order pattern.