A system for students to fill in gaps throughout a math degree

Question

I've noticed that some math majors seem to have arrived at their junior or senior years with some pretty severe gaps in their knowledge of the basics. Ideally, students would be constantly looking back and filling in gaps in their knowledge as they progress through their major. (Probably this is something that the top students do automatically, which is part of the reason for their success.) However, many students seem not to spend enough time or energy on improving their understanding of the basics.

How could the structure of a degree program be modified in order to encourage or require students to keep honing their knowledge of the fundamentals?

Here are a few ideas (not necessarily good ideas, just ideas):

• All math majors could be required to take a cumulative exam at the end of every semester or every year. The cumulative exam would cover everything the students have learned in their math classes up to that point (but focusing on the fundamentals).

• Students could be required to work through the Khan Academy knowledge map and demonstrate proficiency in relevant subjects (as measured by the Khan Academy learning system).

• Students could be enrolled every semester in a 2 unit review class which has homework assignments which review the fundamentals of courses they've already taken.

I'd be interested in hearing any non-traditional ideas for how to structure a degree program along these lines.

Answer 1

1. I think some sort of examination program like qualifier, but for undergrads, would be the simplest way to drive for what you want. Perhaps like the Tripos, albiet not as hard. And with motivation being scoring on it (but hopefully not just in ranking, but in absolute sense).

2. I think you have to consider the disadvantage of spending more time on basics meaning less time on non basics (or even non majors work). Now it may or may not be worthwhile to do so. But you have to analyze trade-offs.

3. You also need to consider that there is a competitive market for students. Kids can go to other schools or to other majors if you make them do too much that is distasteful. So design a solution that seems fun or worthwhile (e.g. for job offers or grad school placement). Or at least that appears or actually does minimize the unpleasant aspects of additional requirements. I guess you could also try to attract by the nature of the demands (like how USMC or SEAL program play on the rigors). But just consider this issue of "selling" the program.

4. It's not clear to me what gaps you are talking about. For instance, not just getting A+ mastery of existing courses? (Do you expect every kid to have mastery?) Curricula that don't include some basics? Is it computational and manipulational mastery that they are missing or something else? More in a particular area (e.g. probability)? Also, some clarity on extent of the gaps (what percent have what gaps, even if just by descriptive estimate of observations but ideally with an assessment). The reason to say this is the more you know about "what the problem is", the more you can address it.

5. I think it's also worthwhile discussing what sort of students and school you are dealing with (Harvard vs UVA vs Cal State vs NW LA State). The nature of the problem/solution may differ a lot.