"I am playing with the children."

Translation: Ich spiele mit den Kindern.

[Dror.Schafer]

Why Kindern and not Kinder?

[Hohenems]

Mit = dative case
Plural of "Kind" in dative is "Kindern"
http://www.canoo.net/inflection/kind:N:N

[slug_a_bed]

does this happen for other nouns too?

[Hohenems]

I'm not a native speaker, but from what I've seen/understood, all nouns are different. There are some general rules (see link), but you just have to learn them as you go along.
http://coerll.utexas.edu/gg/gr/no_03.html

[amonlima]

Why not mit deM Kindern? Isn't Dativ=deM ? This is so hard!!

[Constructive]

You're right, dem is a dative definite article but only for masculine and neuter nouns (singular). In the plural, the dative definite article becomes den.

Ich spiele mit dem Kind. = I play with the child.

Ich spiele mit den Kindern = I play with the children.

Here's a helpful discussion about remembering endings:

https://www.duolingo.com/comment/72?from_skill=bdcb9d10263c7d82f2b5f95ff788fb47

Just remember each mnemonic goes Masculine/Feminine/Neuter/Plural.

[amonlima]

Thank you!