Name of a cognitive test assessing whether a child can think independently I am looking for a names / examples of a tests that determine if the child is thinking on its own (I have tried searching on wikipedia topics about cognitive sciences/ psychology etc. with no luck). To be more specific, long time ago I saw a video that described two tests:

In the first, a teacher was given two cookies, and the kid just one. Teacher asked the child 
"Is it fair that I have two cookies and you just one" and the kid replied as it was supposed to "No". 
Teacher then has split kid's cookie in half and asked again "Is it fair now?" and kid stated "Yes, cause we both have two now". 

My memory fails at accurate description of second test, but I remember that it was supposed to determine if the kid is thinking 'What I know is what everybody knows'.

It's like if you put two kids in the same room with a large amount of 'sth', and asked one, let's call it 'Martha', to leave the room.
Then, you make the kid that stayed take for example two instances of 'sth' and afterwards ask him 'Do you think the kid that left the room knows that you took it?" and kid kid replies "Yes" because it cannot differentiate 'Martha's' knowledge from it's own.

I'd appreciate any ideas where to seek.
The first one is a test if a child has understood conservation of matter. It is an example of a conservation task. These belong to the tests used in the framework of Piaget to test what stage of development a child is in.
Here is a video demonstration of the cookie task. Here is another question on this site pertaining to a different conservation task.

The second one is an example of a false-belief task test for the Theory of Mind. See here for some criticism of the false-belief task, questioning if it truly tests what it aims to test.

I don't think it's appropriate to say either of these is used to test if children are "thinking on their own" for what it's worth. They're about developmental stages.