| Instructors have observed, when teaching junior level courses in abstract algebra, number theory, or real variables, that many students have difficulty out of proportion to the level of difficulty of the material. In an abstract algebra course introducing groups and rings, students' struggles are not affected by the changing of texts, instructors, or the order of presentation. Similarly, experimenting with courses in real variables (say, by treating only functions of a single variable instead of functions on euclidean n-space) offers little relief. The cause of this problem is plain when one considers the previous mathematics courses. The standard calculus sequence is presented, nowadays, to students from various disciplines who have different backgrounds, abilities, and goals, with the aim of teaching them how to differentiate, how to integrate, and how to use these techniques to solve problems. Theorems are stated but usually not proved; hypotheses of theorems are often not verified before applying the theorems (e.g., does one always check whether a given function is continuous?); definitions are given (e.g., limit and convergence) but not taken seriously. After two years of such "mathematics," is it any wonder that a junior-level student is woefully unprepared to read and do real mathematics? |
