I’ve been grading a wheelbarrow-load of papers from my upper-level geometry class this morning. It’s been making me think about the jump from taking calculus to courses beyond calculus. A lot of very good calculus students simply hit the wall when they move on to an "upper-level" course, like linear algebra or geometry. The jump is difficult, I think, because there are certain personality traits that have to be in place for a student to succeed past calculus:
- You have to become very tough-minded. This means you have to begin to be ruthless in your assessment of your own work and the work of others. If you can do better, you have to develop the urge to do so and not be content with cutting your losses on a problem and moving on. Same goes for the work your classmates are doing.
- You have to become self-confident in your mathematical work. In an post-calculus classroom, the correctness of your mathematics is intrinsically, not extrinsically, determined. That means that although there are right and wrong answers out there — and correct and incorrect proofs — the rightness or wrongness is not determined by an authority figure like the back of a book, but rather by the mathematics itself. A proof is correct not because it matches an authority figure’s proof that was published somewhere, but because it meets the standards of logical rigor that a proof requires.
- You must learn to obsess over the right details. It’s easy to avoid obsessing at all, or obsessing over trivialities like grades (yes, I mean that). But it’s difficult to ask the right questions and see the right paths in a problem that need to be taken care of.
Others I’m thinking of are harder to enumerate. For example, it’s easy to do well in calculus if you just learn the system and how to work it. Calculus is usually a pretty straightforward course — you do homework, you take tests, etc. Students who get in a comfort zone in a high school calculus coursde get really offended — perhaps scared — when the college-level analog of that course asks them to do more than just calculate derivatives. Likewise, a lot of students decide they want to study mathematics because they figure it’s a system they learned how to play and can continue to play until they get a degree. Usually if you ask, they’ll say they got into math because "it was always easy" and "there’s only one right answer". But pretty quickly after calculus, students become very exposed in their thinking because there is no longer such an emphasis on plug-n-chug calculating.
Additionally, I think a lot of education majors who end up taking these upper-level math courses have a hard time because many of the characteristics of a successful upper-level math student — described by adjectives like tough, demanding, self-confident, meticulous, etc. — sound like the direct opposite of the characteristics of a successful K-12 teacher as advertised by many education schools. The ed schools seem to want teachers who are nurturing, caring, etc. and it’s very hard — certainly not often made clear — that a teacher can, and indeed must, be simultaneously caring and tough, nurturing and demanding, strongly self-confident and open to correction, and so on. The assumption is that the nurturing/caring and the tough/demanding groups of characteristics are mutually exclusive, and the former trumps the latter. It’s a rare ed school that stresses the centrality of the fact that it’s not the teacher’s job to be liked.