"I want to show that I can count."

Translation: Chcę pokazać, że umiem liczyć.

[IrakliusDaCosta]

So "mogę" instead of "umiem" won't work here?

[immery]

It is about the ability you have learned, so "umiem"

[Walkinthedog]

Some nights I should not be doing this, this is one of them.

[Garry_S]

Do "rachować" and "liczyć" mean the same?

[alik1989]

Rachować means to calculate, to do calculations, to do (the) math(s). The dictionary refers to it as 'dated'.