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…