Learn Hardcore German: Die Kinder sitzen am Brunnen und ich sehe sie Steine ins Wasser werfen. - The children sit by the fountain and I see them throw stones into the water.

an + Dativ is used to express being at / by / next to a vertical surface or an object like a wall, door, fountain, etc.
im is a contraction of in dem, and in + Dativ usually means being inside something.

So:

am Brunnen = an dem Brunnen = at/by the fountain (next to it)
im Brunnen = in the fountain (inside it, probably in the water)

The sentence wants to say they are sitting at the fountain, not inside it, so am Brunnen is correct.
The full, non‑contracted form an dem Brunnen is technically correct but sounds overly formal here; in everyday speech and writing people almost always say am Brunnen.

Why is there no comma before und ich sehe?

German allows (but does not require) a comma between two main clauses connected by und or oder.

Here we have two main clauses:

Die Kinder sitzen am Brunnen
ich sehe sie Steine ins Wasser werfen

They are joined by und:

Die Kinder sitzen am Brunnen und ich sehe sie Steine ins Wasser werfen.

According to modern rules, both of these are correct:

Die Kinder sitzen am Brunnen und ich sehe sie Steine ins Wasser werfen. (no comma)
Die Kinder sitzen am Brunnen, und ich sehe sie Steine ins Wasser werfen. (comma)

The version without the comma is more common in everyday writing; the comma can be added if the sentence is long or complex and you want to make the structure clearer.

Why is it ich sehe sie Steine ins Wasser werfen and not ich sehe, dass sie Steine ins Wasser werfen?

Both versions are possible, but they use different constructions:

ich sehe sie Steine ins Wasser werfen
This is an AcI structure (Accusativus cum Infinitivo), common with verbs of perception (sehen, hören, fühlen etc.). The pattern is:
- ich sehe (finite verb)
- sie (accusative object = the people I see)
- Steine ins Wasser werfen (infinitive phrase describing what they are doing)
Literally: I see them throw stones into the water.
ich sehe, dass sie Steine ins Wasser werfen
Here dass sie Steine ins Wasser werfen is a subordinate dass-clause.
Literally: I see that they are throwing stones into the water.

Stylistically, the AcI version (ich sehe sie … werfen) is shorter and sounds a bit more immediate/visual. The dass version is slightly more abstract or explanatory. Both are grammatically correct; the sentence just happens to use the perception‑verb construction.

Why is the verb werfen at the very end of the sentence?

In ich sehe sie Steine ins Wasser werfen, the verb werfen is an infinitive that belongs to the perception‑verb construction:

Main-clause finite verb: sehe (in second position)
Infinitive at the end: werfen

German word order rules for clauses with an infinitive say that the infinitive goes to the end of the clause. Everything that belongs to that infinitive (its object and adverbials) normally stands before it:

sie (object of sehe, and also the doer of werfen)
Steine (object of werfen)
ins Wasser (directional phrase with werfen)
werfen (infinitive, at the end)

So the order is:

ich – sehe – sie – Steine – ins Wasser – werfen

You cannot say:

*ich sehe sie werfen Steine ins Wasser (wrong word order)
*ich sehe sie Steine werfen ins Wasser (also wrong)

Why is it sie and not Sie? How do I know what sie refers to?

Sie with a capital S is the formal you (singular or plural).
sie with a small s can be:

she (singular, nominative)
they (plural, nominative)
her (singular, accusative)
them (plural, accusative)

Here, sie refers back to die Kinder:

Die Kinder is plural.
In ich sehe sie …, sie is the accusative object of sehe.
So sie here means them = the children.

You can check this by replacing it:

Ich sehe die Kinder Steine ins Wasser werfen.
→ That works and has the same meaning, so sie must stand for die Kinder.

Why is Steine used without an article? Why not die Steine?

German often omits the article with indefinite plural nouns, similar to English:

Kinder spielen im Garten. = Children are playing in the garden.
Wir kaufen Äpfel. = We are buying apples.

Steine here means some stones, not specific, known stones:

sie werfen Steine ins Wasser = they are throwing stones into the water

If the stones were specific (for example, stones we already talked about), you could say:

Ich sehe sie die Steine ins Wasser werfen.
→ I see them throw the stones into the water.

But that would change the meaning to those particular stones. The original sentence just talks about stones in general, so no article is normal and idiomatic.

What does ins Wasser mean exactly, and why is it not im Wasser?

ins is a contraction of in das (preposition in + accusative article das).

in + Akkusativ = movement into something (direction)
in + Dativ = being inside something (location)

So:

ins Wasser = in das Wasser = into the water (movement)
im Wasser = in dem Wasser = in the water (location, no movement)

Because the children are throwing stones, we have a movement towards a goal, so we use in + Akkusativ:

Steine ins Wasser werfen = to throw stones into the water

What cases are used in this sentence (die Kinder, am Brunnen, sie, Steine, ins Wasser)?

Breakdown by case:

Die Kinder – Nominative plural
Subject of the first clause: The children are sitting…
am Brunnen = an dem Brunnen – Dative singular
an + Dativ for location: at/by the fountain
ich – Nominative singular
Subject of the second clause: I see…
sie – Accusative plural
Direct object of sehe: I see them… (refers to die Kinder)
Steine – Accusative plural
Direct object of werfen: throw stones…
ins Wasser = in das Wasser – Accusative singular
Direction towards something with in + Akkusativ: into the water

Why is it sitzen and not something like sind sitzen for “are sitting”?

German does not have a regular progressive tense (like English am sitting, are throwing) formed with sein + -ing.

Instead, German normally just uses the simple present:

Die Kinder sitzen am Brunnen.
= The children sit at the fountain.
or in natural English: The children are sitting at the fountain.

The form sind sitzen does not exist in standard German. To express a continuous action, German simply uses the present tense:

Sie lesen. = They are reading.
Er arbeitet. = He is working.

So sitzen here is the normal present form, and it already covers the meaning are sitting.

What is the difference between sitzen and setzen?

They describe two different kinds of action/state:

sitzen – to be sitting (state, no movement)
- Die Kinder sitzen am Brunnen.
  = The children are sitting at the fountain.
(sich) setzen – to sit down / to place oneself (movement into the sitting position)
- Die Kinder setzen sich an den Brunnen.
  = The children sit down at the fountain.

So:

Use sitzen when you are talking about the position someone is already in.
Use (sich) setzen when you are talking about the movement of sitting down.

Why is the tense simple present everywhere? Could we also say something like “while” in German?

German uses the simple present for many things where English prefers the present progressive:

Die Kinder sitzen am Brunnen → The children are sitting at the fountain.
ich sehe sie Steine ins Wasser werfen → I see them throwing stones into the water.

If you want to explicitly express “while” in German, you can use während:

Während die Kinder am Brunnen sitzen, sehe ich sie Steine ins Wasser werfen.
= While the children are sitting at the fountain, I see them throwing stones into the water.

But it’s not necessary. The original sentence already clearly describes two things happening at the same time, and German is very happy with just the simple present for that.

Could we also say Die Kinder sitzen beim Brunnen? What is the difference from am Brunnen?

Yes, Die Kinder sitzen beim Brunnen is also grammatically correct.

am Brunnen = an dem Brunnen – literally at/by the fountain
Focuses a bit more on being directly next to it, in contact with its area.
beim Brunnen = bei dem Brunnen – by/near the fountain
bei often suggests being in the vicinity of something, not necessarily touching it or right up against it.

The difference is small and context‑dependent, and in many situations they can be used almost interchangeably. Here, am Brunnen is slightly more idiomatic and vivid, as if they are really sitting right at the fountain’s edge.