I know an undergraduate who just finished calc III, and really struggled with it. However, it seems like a lot of the issues are coming from skills which should have been established during algebra, precalc, and calc I. I want to recommend some resources (practice problems, videos, etc.) to help him improve at these skills, does anyone have any recommendations?
Here are some examples to illustrate the sort of skills I'm hoping he can practice:
- As part of a larger problem, he had to solve cos(x) = 0.4, and wasn't able to apply the idea an inverse function (even after being reminded about the existence of inverse-cosine)
- Often when talking through a problem, he'll struggle with the fact that he was introduced to a formula using different letters. E.g., when learning how to compute a directional derivative they used v for the vector, but when needing to calculate a directional derivative using a new vector which was called p, he was confused. This issue in particular has come up in across a ton of different problems I've helped him with.
- Generally connecting the math to the actual situation being described is very hard for him. This leads to a lot of trouble when problems are given using language which does not precisely match the language which the methods were originally presented using. This in particular seems like a hard thing to study without tutoring/instructor feedback.
- This may be related to the last point, but when problems are given which flip "usual" question/answer information, the idea of writing out the known formulas and solving for the requested information is hard to prompt him to come up with. For example, when given a vector v in 2-D and told "find any vector orthogonal to v", the idea of setting the dot product of v and the unknown vector equal to 0 did not come to mind, even after we talked about what how we know when two vectors are orthogonal.
I know this is a lot, but he's hoping to apply to PhD programs in a pretty technical field, so these are things he'll need to catch up to his cohort on. Does anyone have any advice on self-study material? (Other advice about how to make progress on improving would also be helpful)
P.S. One last meta-issue which may be particularly pertinent to self-studying is that he seems to have a lot of trouble knowing when he does or doesn't know something. When I've self-taught math subjects I've generally found that I'm able to identify when I'm not understanding something and seek outside help. Does anyone have any advice on how to build this ability up?