Treating Mathematics as an Ill-Structured Discipline

Educators and cognitive scientists commonly think of mathematics as the paradigmatic “well-structured discipline.” Mathematics is regarded as a field in which statements have unambiguous meanings, there is a clear hierarchy of knowledge, and the range of possible actions in response to any problem is both restricted and well defined in advance. Cognitive scientists frequently contrast mathematics and mathematical logic problems—which must and can be solved within the narrow constraints of accepted postulates and transformations—with problems in such domains as the social sciences, where large amounts of external knowledge must be brought to bear, texts draw alternative interpretations, and conclusions can be defended rationally but not always strictly proven. Educators typically treat mathematics as a field with no open questions and no arguments, at least none that young students or those not particularly talented in mathematics can appreciate. Consider in evidence how little discussion occurs in typical mathematics classrooms compared with English, social studies, and some science classrooms. Even when we teach problem solving, we often present stereotyped problems and look for rules that students can use to decide what the right interpretation of the problem is—so that they can find the single appropriate answer.