Learn Hardcore Dutch: Op school kijkt zij ook huiswerk na, en dat doet zij volledig vrijwillig. - At school she also checks homework, and she does that completely voluntarily.

Dutch has a verb-second rule in main clauses: the finite verb (here kijkt) must be in second position, but many different things can come in first position.

In Op school kijkt zij ook huiswerk na, the first position is taken by the adverbial phrase Op school, so kijkt must come second and zij is pushed after the verb.
If you start with the subject, you get Zij kijkt op school ook huiswerk na, which is also correct.

So you can say:

Op school kijkt zij ook huiswerk na. – Focus on the place (at school).
Zij kijkt op school ook huiswerk na. – More neutral, or focus on she.

Both are grammatical; the choice is about emphasis and style.

Why is it kijkt zij … na instead of just kijkt zij huiswerk? What is na doing here?

Nakijken is a separable verb in Dutch:

Base form: nakijken – to check / correct (e.g. homework, tests)
Separable in main clauses: kijken … na

In the sentence:

kijkt = finite verb in 2nd position
na = separable particle pushed to (near) the end of the clause

So:

Zij kijkt het huiswerk na. – She corrects the homework.
In perfect tense: Zij heeft het huiswerk nagekeken. (The parts are joined again.)

You cannot say Zij kijkt huiswerk if you mean she corrects homework; that sounds like she looks at homework. You need nakijken for the “to check/correct” meaning.

What is the difference between zij and ze here? Could I say Op school kijkt ze ook huiswerk na?

Yes, you can say Op school kijkt ze ook huiswerk na; it is perfectly normal.

Differences:

zij – full/stressed form, used:
- when you want emphasis on the subject (zij = she, not someone else), or
- in more formal or careful speech/writing.
ze – reduced/unstressed form, very common in everyday speech when there is no special emphasis.

In … en dat doet zij volledig vrijwillig, using zij instead of ze adds a bit of emphasis: and she does that entirely voluntarily (no one makes her). In casual spoken Dutch you would often hear … en dat doet ze volledig vrijwillig.

Why is ook placed before huiswerk? Could it go somewhere else, like Op school kijkt zij huiswerk ook na?

Ook in Dutch usually appears before the element it modifies or emphasizes. In this sentence, the most natural and common slot is:

Op school kijkt zij ook huiswerk na.
– At school she also checks homework (in addition to other things).

Other positions:

Op school kijkt zij ook huiswerk na.
- Neutral and most common.
Op school kijkt zij huiswerk ook na.
- Possible, but sounds less natural and can sound as if you are contrasting huiswerk with something else she might or might not check.
Ook op school kijkt zij huiswerk na.
- Emphasis on op school (also at school she checks homework, maybe in addition to checking homework elsewhere).

So: the given position of ook is the most idiomatic for the intended meaning.

Why is there no article before huiswerk? Why not het huiswerk?

Huiswerk usually behaves like an uncountable / mass noun in Dutch, similar to homework in English.

Without article (very common):
- Zij kijkt huiswerk na. – She corrects homework (in general; some homework).
With het (more specific):
- Zij kijkt het huiswerk na. – She corrects the homework (a particular set known from context, e.g. the homework from yesterday).

In your sentence, huiswerk refers to homework in general that she checks at school, not to a specific one instance, so dropping the article is natural.

Using de huiswerken (plural) is rare and usually only in very specific institutional or formal contexts.

What does dat stand for in en dat doet zij volledig vrijwillig? Why is it dat and not dit?

Here dat is a demonstrative pronoun referring back to the whole previous action:

dat = that (thing / action)
→ That, she does entirely voluntarily.

So dat represents the fact that she corrects homework at school.

Why not dit?

dat is normally used to refer back to something just mentioned (especially in writing or neutral style).
dit is used more for something near or being introduced now, often with a bit more immediacy or “closeness.”

In this context, dat is the default choice. Dit would sound odd or marked unless you had a special discourse reason.

What is the function of doet in dat doet zij volledig vrijwillig? Could I just say en dat is volledig vrijwillig?

Doet is a pro-verb here: it stands in for the full verb phrase huiswerk nakijken.

Full idea: Zij kijkt huiswerk na, en zij doet dat volledig vrijwillig.
→ She corrects homework, and she does that entirely voluntarily.

The actual order is just flipped for style and emphasis: … en dat doet zij volledig vrijwillig.

Could you say en dat is volledig vrijwillig?
Yes, that is also grammatical and understandable:

… en dat is volledig vrijwillig. – and that is completely voluntary.

However:

dat doet zij volledig vrijwillig emphasizes her action (she is the one acting voluntarily).
dat is volledig vrijwillig emphasizes the status of the activity as voluntary.

Both are fine, but the original keeps the focus on her doing it.

How does volledig vrijwillig work grammatically? Is vrijwillig an adjective or an adverb here?

In Dutch, many words that are adjectives can also function as adverbs without changing form, and vrijwillig is one of them.

As adjective:
- een vrijwillige taak – a voluntary task
As adverb (describing how she does it):
- Zij doet dat vrijwillig. – She does that voluntarily.

In dat doet zij volledig vrijwillig:

volledig (completely) modifies vrijwillig (voluntary/voluntarily),
the pair volledig vrijwillig functions adverbially, describing how she does it.

Paraphrase: Zij doet dat op een volledig vrijwillige manier.

Why is there a comma before en in …, en dat doet zij volledig vrijwillig? Is that required?

In Dutch, a comma before en is optional when joining two main clauses. It is often used when:

the clauses are relatively long, or
you want to mark a slight pause or change in focus.

Here we have two full main clauses:

Op school kijkt zij ook huiswerk na
dat doet zij volledig vrijwillig

They share the same subject (zij, though expressed differently), but structurally each could be a separate sentence. So:

Op school kijkt zij ook huiswerk na en dat doet zij volledig vrijwillig.
Op school kijkt zij ook huiswerk na, en dat doet zij volledig vrijwillig.

Both are correct. The comma simply adds a small pause and makes the structure a bit clearer in writing.

Why is it Op school and not In school, like in English at school vs in school?

Prepositions differ between languages; you cannot always translate them literally.

Dutch typically uses op school to mean at school (as a place where you study/work).
in de school would be more literal inside the building of the school and is used much less often in this abstract sense.

Roughly:

op school – at school (as an institution / regular location)
in de school – in the school building (physical interior), when that literal meaning matters

So Op school kijkt zij ook huiswerk na corresponds to English At school she also corrects homework.

Is the word order kijkt zij the same as in a question like Kijkt zij ook huiswerk na? Why isn’t this a question?

The sequence kijkt zij appears both in statements and yes/no questions, but the position of the first element is different:

Statement with another element in first position:
- Op school kijkt zij ook huiswerk na.
  - First position: Op school
  - Second: kijkt (finite verb)
  - Then: zij
Yes/no question:
- Kijkt zij ook huiswerk na?
  - First position: kijkt (finite verb)
  - Second: zij

So:

Verb-first = yes/no question
Something-else-first, verb-second = normal main-clause statement

In writing, the question mark and context also signal whether it’s a question or a statement.