Why do we see the sets in this order? There is the following question: 
Why do we see in maths first the set of integers $\mathbb{Z}$ and then the set of rational numbers $\mathbb{Q}$ but in school we see first the rationals and the negative numbers? 
Do you have an idea why it is like that?
 A: The reason why, in mathematics, the set of integers is seen first is : the rationals are defined in terms of equivalence relation on the set of integers. 
So, the reason is contained in the construction process of number sets. 
See : Peterson, Theory Of Arithmetics ( at archive.org). 
Now, why do not school mathematics follow this order? 
There was a time during which,  it was thought that school mathematics had to undergo a huge reform, in order to follow the scientific order of Modern Mathematics. 
So, pupils were taught at 11 or 12 years old : sets, equivalence relations, orderings, etc. 
But this led to a failure. ( https://en.wikipedia.org/wiki/New_Math)
The experience of " New mathematics" (or " Modern mathematics"), showed that the order in which mathematics are learned, is not the same as the order in which  mathematical theories actually develop. 
The process of learning, is not the same as the process by which mathematical objects are constructed. 
An analogy: scientifically speaking, newtonian mechanics comes after einsteinian mechanics, since the newtonian one, is a particular case relatively speaking, to the general theory of Einstein. However, at school, newtonian mechanics is seen first.
