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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?

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    $\begingroup$ He sounds like someone who has memorized everything, and never tried to actually reason with it. I'd prescribe puzzles, actually. $\endgroup$
    – Sue VanHattum
    Dec 19, 2022 at 5:13
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    $\begingroup$ he's hoping to apply to PhD programs in a pretty technical field --- I hate to rain on this person's parade, but someone with aspirations this high (even if still only a 2nd or 3rd year undergraduate) should be well on their way to knowing how to take care of things like this by theirself (or realize that a PhD is not a reasonable goal). I'm not saying they should already know where to look for the kind of help they need, but they should know something about the kind of help they need and they should know HOW to go about learning where to look for help. (continued) $\endgroup$ Dec 19, 2022 at 15:57
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    $\begingroup$ Based on what you've written, there seems to be a HUGE gap in this regard (being able to learn things independently, and being pro-active at removing deficiencies) between where such a person seems to be at now and where they need to be at when taking no-nonsense and fast-paced graduate courses, passing their Ph.D. qualifying exams, writing a Ph.D. dissertation, etc. $\endgroup$ Dec 19, 2022 at 15:57
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    $\begingroup$ Agree with @DaveLRenfro. Inverse trig functions are covered in secondary school usually around age 16. Formulas with different letters is algebra 1, usually around age 14. Connecting formulas to word problems generally starts in upper primary, around age 10-12. I'm being generous with these ages, because math enthusiasts probably learned these topics earlier than this. How has this person passed Calc III? $\endgroup$
    – shoover
    Dec 19, 2022 at 17:46
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    $\begingroup$ @shoover: To clarify, my point is not so much that the student should have learned this material much earlier (although they should have) as it is that the student should be better at self-assessment and self-remediation by this point. For example, the OP wrote: 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. This also likely applies to most anything replacing "math subjects", although there might not be the desire or the need to do so, and this should be the same for any Ph.D. aspirant. $\endgroup$ Dec 19, 2022 at 18:55

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I am sceptical whether just doing more exercises would help this student. Based on the description, they are thinking about mathematics the wrong way.

The problematic skill seems to be sense-making, ie getting at the meaning behind the mathematical formalism and actually reason about things. While it is very common for beginners to struggle with that, if they haven't gotten it by Calc III, it seems like a cause for real concern.

Unless the additional concreteness of meaning in Microeconomics makes a difference to their sense-making ability, going into a PhD in that field without fixing this seems like a horrible idea - you can solve exercises by blindly following steps you've learned by heart, but you can't do research like that.

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  • $\begingroup$ First, the student is struggling with basic problems he has seen before. Secondly, weaker students have lower sense making and need more drill. Thirdly there is no magic switch to turn on sense making. Fourthly, we need to know more about the students grad program to know if he needs super msth skills, if he's ever going to hack it, or even what his other gaps are. But not everyone needs to be Tukey. $\endgroup$
    – guest
    Dec 19, 2022 at 18:55
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    $\begingroup$ @Guest - you seem to think that being able to solve these exercises is actually valuable in itself. It isn't. The only goal is sense making, and the exercises are a means to the end of sense making, not an end in themselves. $\endgroup$ Dec 20, 2022 at 19:05
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Sense-making versus doing lots of exercises. Both are important. But this particular student seems to really not get that math is sense-making.

I am guessing you are his tutor. Would you like to ask him to play with the Rush Hour puzzle for a few minutes at the beginning of a session? There are lots more fun and challenging logic puzzles on that site.

Desmos has oodles of good activities. One I have used extensively is marbleslides. It's meant to help students think about equations versus graphs. You didn't say that he was bad at that, but if he's just memorized rules, this will help him build a more robust understanding. (I linked the one for straight lines; there is a whole collection for different function types, and more.)

Catriona Schearer makes wonderful (hard) puzzles.

All of these seem like they focus somewhat on spatial reasoning. Hmm. Set is good too, logic and not spatial.

I can't think of a good game specifially for notation. I suppose you could create one yourself. Like concentration. Have a bunch of cards that all mean the same thing, but using different symbols and arranged in algebraically different ways. (My students often have trouble recognizing that $\sqrt{9-{x^{2}}}$ is (the upper half of) a circle. They would benefit from a game like this.)

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    $\begingroup$ Thanks for the suggestions. As for the tutor thing I'm just a friend of his with no professional relationship (also I don't even attend his institution) $\endgroup$ Dec 20, 2022 at 7:02
  • $\begingroup$ You mentioned Catriona Schearer. This is her twitter account. On her pin tweet, she has put a link to the google-docs document you mentioned. $\endgroup$ Dec 20, 2022 at 16:43
  • $\begingroup$ I was avoiding twitter. I guess until she leaves, that's the source. I have now edited my answer. Thank you. $\endgroup$
    – Sue VanHattum
    Dec 20, 2022 at 19:54
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Two somewaht different thoughts.

  1. In terms of drill volume, I like Frank Ayres Schaum's Outline First Year College Math (pre-calculus), the original edition. Has the answers. Goal is just to do lots of volume. No sudden "aha", just more facility from exposure. There are competing "huge problem" books. Also online solutions like Kahn Academy. But the one I mentioned is one I know that still gets good reviews on Amazon.

  2. I would also suggest, for efficiency, just doing drill in the context of his degree (using the familiar variable names), which you did not mention. Trying to guess one that goes through calc 3, but not ODEs. Thinking comp sci or chemistry. Realistically, remediating his general math skills may not be as efficient as just brushing him up on the specifics of his major (e.g. "p chem"). With adult learners, moved on to grad school, sometimes you just have to be more economical. Like I really could care less if the average coder or natural products synthetic chemist knows div grad curl. Or even arccosine.

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  • $\begingroup$ He's doing economics, so he'll need to take analysis I believe (not 100% sure, I'm a CS PhD student so I'm lacking the econ specific context). $\endgroup$ Dec 19, 2022 at 2:18
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    $\begingroup$ As for the arccos comment, it's more the idea of manipulating function inverses (or even reasoning about unknown functions) which I'm concerned about $\endgroup$ Dec 19, 2022 at 2:18
  • $\begingroup$ Re first comment: knowledge is power. Would be helpful to know more about the type of program he is going into and how important the math really is. Econ can vary a lot. $\endgroup$ Dec 19, 2022 at 2:24
  • $\begingroup$ Re second comment, you have to pick your battles. He needs to learn his econ also. Being super smart at concepts of functions may not be critical. But even here, I think just more sheer exposure (volume) to basic math and manipulations would help him, as opposed to some concept knowledge "flip". I don't have to think about "inverse" land if I've just done a shlew of back and forth with trig functions. $\endgroup$ Dec 19, 2022 at 2:27
  • $\begingroup$ To both points that's definitely true, but I do feel like quite a bit of this underlies quantitative and symbolic reasoning in general. I've had the "this is using different letters" problem while walking him through a micro-econ problem as well. I agree with you that volume will help so I'll definitely point him to this book. $\endgroup$ Dec 19, 2022 at 2:31

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