Why mathematics teachers teach non-mathematical stuffs?

Recently, I gave tuition to secondary 1 students, and I think nowadays mathematics teachers are skipping steps, while last time only students skip steps.

For example, 3x + 6 = 18, and the next line we put 3x = 18 - 6.
Mathematically, we subtracted 6 from both sides because we want to single out the variable x.
However, this is usually not being taught, or it was just mentioned without any intention of teaching by assuming that students won't understand. As a results, I saw in student's workings this: above the first equation, there is an arrow from above the number 6 going across to the right side of the number 18. That is, what students have learnt was they just "move"/"throw"/"bring" the number 6 over to the other side, and "change"/"convert" the plus sign to a negative sign. What is this!!! Learning mathematics, or learning magical or pattern rules of manipulating symbols?

There are many more such non-mathematical stuffs that are taught in schools!