Spanish language Q&A Forum

Like Ian, I am confused by some of the examples given here, in particular:

Iremos adonde tú quieras, cariño >> We will go where (to whichever restaurant) you want, darling...

... which does involve movement, but despite that it uses 'donde' - with no accent !

And in the short quiz included in this site, I got this question wrong:

We are taking the children to where they can run around freely > Llevamos a los niños ________ puedan corretear libremente…. i.e., I [incorrectly !] chose "adónde" with an accent because it did seem to involve movement … (but no ! - it had to be "adonde").

These seem to have similar constructions for similar meanings (interim completion of an action that may continue). Is there a distinction between them I'm not getting? 

Why is cambiar used with the preposition de and sometimes without? Is there any difference between the two ways of using cambiar?

"Mathematics" can be taken as a singular noun. Therefore it seems that a correct format in Spanish would be 

"te gusta mathematics".