They do teach rote memorization in public schools. However, they are cluttering brains with nonsense at the same time. I am not sure if the clutter is why there are a lot of students in fourth grade who still do not know their multiplication tables. Last year, third graders were not tested on multiplication facts until well into December if not later. This year, they are pushing multiplication facts earlier though they continue to test students on basic addition and subtraction facts via timed tests. The first graders are starting to focus on addition facts this week. The clutter is the problem!
They are hereabouts trying to teach all sorts of convoluted ways to approximate answers with no emphasis on whether the answers are right or wrong. They older teachers are still having the kids memorize multiplication tables and addition facts (in younger grades) but the practice is discouraged, though not yet forbidden, because it is seen as making thinking processes rigid. They think they are teaching kids how to think.