If you ask an average person on the street what is the highest level of mathematics, the most common answer would probably be Calculus. There might even be a few throwing College Algebra out there as fairly advance. However, if you ask a math major or engineering student the question of what is the lowest level of mathematics, the foundation of the mathematics they use, the most common answer would likely be Calculus. Why the disparity?
The mathematics that is taught from kindergarten through secondary is often limited to procedural techniques to solve specific problem types without recognizing that advanced mathematics is all about recognizing patterns and using axioms, definitions, and theorems to formalize those patterns thereby leading to new patterns.
So, of course, in upper level mathematics we spend a great deal of time moving from procedural mathematics, to proving theorems, to developing new theorems. Unfortunately, the rigor is sometimes lost in the classroom for many reasons. I’m sure I’ve used just about every one of the following invalid proof techniques.