Beliefs about how to do mathematics and what it means to know it in school are acquired through years of watching, listening, and practicing. Mathematics problems have one and only one right answer. Mathematics is a solitary activity, done by individuals in isolation. Students who have understood the mathematics they have studied will be able to solve any assigned problem in five minutes or less. The mathematics learned in school has little or nothing to do with the real world.
Formal proof is irrelevant to processes of discovery or invention.
The basic arguments regarding student beliefs were made in part I. As an illustration, we point to the genesis and consequences of one belief, regarding the amount of time students believe that it is appropriate to spend working mathematics problems. The data come from year-long observations of high school geometry classes. Over the period of a full school year, none of the students in any of the dozen classes we observed worked mathematical tasks that could seriously be called problems.