I taught my first math class in 1965. I taught my last class in 2017. A disturbing transformation has occurred in those 52 years. In those early years of my career we taught concepts, generalizations, abstractions, deductive reasoning, and critical thinking (although Richard Paul had not yet invented the term). The same things were taught some years prior to that when I was a student.

The learning objectives were the concepts, generalizations, abstractions, deductive reasoning, and critical thinking. We used examples and exercises merely to illustrate and clarify individual instances of concepts, generalizations, abstractions, deductive reasoning, and critical thinking.

With the passage of time the examples and exercises themselves have become the learning objectives. Concepts, generalizations, abstractions, deductive reasoning, and critical thinking have disappeared from math classes and we merely train (as opposed to teach) students to “get the answer” to inane textbook problems.

These inane textbook problems are not realistic, and every student correctly knows that he does not need the goofy memorized tricks to “get the answer” to a problem that will never happen outside the classroom.

The part of mathematics which is useful in a very real way is concepts, generalizations, abstractions, deductive reasoning, and critical thinking. Those are nowhere to be found in the majority of mathematics classroom of 2018.