Grant Wiggins shared an article on his blog called "The Nature of Proof." The article describes a course in geometry given in the 1930s that was not only extremely influential for those who took the course, many of them described it as the most important course they ever took in their entire lives. Here’s a quote from the teacher of that course.
"While teachers of mathematics say they want the young people in our secondary schools to understand the nature of proof, that should not be and probably is not their total concern. What these teachers really want is not only that these young people should understand the nature of proof but that their way of life should show that they understand it. Of what value is it for a pupil to understand thoroughly what a proof means if it does not clarify his thinking and make him more "critical of new ideas presented"? [emphasis mine] The real value of this sort of training to any pupils id determined by it effect on his behavior and for purposes of this study we shall assume that if he clearly understands these aspects of the nature of proof his behavior will be marked by the following characteristics:
- He will select the significant words and phrases in any statement that is important to him and ask that they be carefully defined.
- He will require evidence in support of any conclusion he is pressed to accept.
- He will analyze the evidence and distinguish fact from assumption.
- He will recognize stated and unstated assumptions essential to the conclusion.
- He will evaluate these assumptions, accepting some and rejecting others.
- He will evaluate the argument, accepting or rejecting the conclusion.
- He will constantly re-examine the assumptions which are behind his beliefs and which guide his actions."
Harold Fawcett, The Nature of Proof, Page 11, 1938
The use of "He" in this quote should be understood in the context of when this quote was written, and could easily be substituted for "The student" or something similar.