An examination of college mathematics majors’ understandings of their own written definitions

• Students’ written definitions are poor descriptors of their concept images.
• Students’ definitions frequently conceal surprising and faulty concept images.
• Students fail to consider key examples when writing definitions.
• Students have vague concept images for concepts in question and of related concepts.
• Recommend that teachers not rely on definition questions on exams to assess concept images.

This qualitative study of ten undergraduate mathematics majors examined students’ abilities to write definitions and found that students at the advanced level of undergraduate mathematical study have difficulty creating definitions that conform to their concept images or to accepted definitions of basic concepts. This is due in part to (1) failure to consider key examples when writing definitions, (2) weak concept images for the concept in question, and (3) vague concept images for related concepts. The results of this study have implications for secondary-level and college-level mathematics instruction.