Mathematics has been overly influenced by outdated social structures. In A Mathematician's Lament, Paul Lockhart rightfully refers to mathematics as an "empty shell." Mathematics is empty because students all over graduate high school and university without having one meaningful experience. Textbooks are only contributing to the problem. Students are being denied the opportunity to make mistakes, to make discoveries, and to experience creative frustration. There is no time wasted when doing mathematics. The current system's path of least resistance creates schools devoid of any human experiences or mathematics for that matter. We do not teach students to remind themselves of where they want to be in the end when they are stuck.
In my discrete mathematics class at UT Austin, my professor taught us all choices have consequences. When doing math, we have to care. In a class on rings, we discussed 1+1. The biggest lesson turned out not to be what 1+1 is, but what we define 1+1 to be. For instance, if we restricted the range to 0 and 1, would the answer be 0? Or would the answer be 1?
What makes people fully human is allowing them to ask questions like these for themselves in solitude and arrive at a truth. As a culture, we have stripped beauty, inspiration, and art in mathematics away from students. Education as not about placing knowledge inside the soul similar to one putting sight into the eyes of someone who was blind. In The Allegory of the Cave, Plato stressed the soul possesses that knowledge already.
The knowledge of how the soul can most easily and most effectively be turned around. What our current education system needs is "notions" not "notations." We do not need to treat students like "trained chimpanzee" with overhead projectors and printed notes during lecture.
The capacity to learn mathematics is present in everyone. Education is the art of turning around, as the soul may not be turned the right way or looking where it should be. Plato stated, "This is what education has to deal with."