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This is a little bit of a niche topic.

I've dealt with a pretty bad dose of long COVID that has caused some serious gaps in my mathematics (basically causing terrible arithmetic skills and a really shaky foundation). Here's essentially what I'm dealing with. I've seen the mathematics in old courses (Ross' Analysis, etc.) before but my memory is shaky and when I go to do problems I know how to start but my newfound significant lack of algebra,trigonometry,etc. hinders me from properly completing the proof. Therefore, I know I need to go back and review from the ground up before relearning the necessary proof based math foundations like Abstract Algebra, Analysis, Topology, etc.

I am debating whether to go down the path of using the Beast Academy and Art of Problem Solving books to do so or going through books used in Developmental Mathematics courses and then following the "conventional" community college/university path that leads to Stewart's Calculus, etc.

For someone wishing to go back to grad school in Math, which route would you take and why?

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    $\begingroup$ Bummer! Sorry to hear about that. This is a new issue. I'm guessing none of us know well enough what happened to your brain to have good advice. I'm guessing Beast and AOPS would be way more fun than the other books. Alcumus on AOPS is free, so that's a help. $\endgroup$
    – Sue VanHattum
    Jan 21, 2023 at 0:48

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I would like to complement existing answers by suggesting spaced repetition. I have been applying it myself and have found myself feeling like I had newfound powers.

I use a free app called Mochi. There is a mobile version but I use it on my notebook. You can start with simple flashcards but it can do so much more: you can use cloze deletions (also called occlusion tests) on both text and images (by using the diagram card). It accepts LaTeX so you can input formulas easily, both inline and as blocks.

It is also a good method for learning because it keeps you actively thinking about the material in order to build your cards. Ever since I've started using this app feels like my skills have gone up drastically, merely because I've stopped struggling with a full working memory and instead began to rely on a well-honed long-term memory. I've written more extensively about working memory and long-term memory in this answer, if you're interested in learning more.

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If you have gaps, Khan Academy is a great way to go. It has videos teaching topics that you've forgotten, lots of practice material, and it will keep track of what you've mastered and what you need more help with. It will assign problems for your weaknesses.

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  1. If you want to go down this path: I would go with a review text instead, like Frank Ayres First Year College Math. (Schaum's Outline, I'm familiar with the first edition, available for cheaps on the Intertubes.) I have a strong feeling that if you work all the problems--you do know you need to work problems, right?--you will get enough review of algebra 1/2 (US meaning), Euclidean geometry, pre-calc, trig, analytic geometry. In the process, you should get enough basic repetition of arithmetic also (especially fractions). I think reviewing long division and percentages from the ground up, would be a waste of time. I think doing an entire curriculum like AOPS would be overkills and not efficient as it is designed for new learners. Bright ones, but still new learners.

  2. As far as your other implied question, "I am debating", my advice would be NOT to do the math grad school. It is a much better degree for the top performers, but has very low economic return for the lower end. I fear that you would be avoiding moving on with your life. In comp sci or business or some other field, you'd be quite competitive.

    [This is only partially to do with the Covid...I completely believe you on the brain fog and the like. Had a buddy who dealt with it. And have had my own issues with POTS. But researching your previous questions, I'm not seeing a heavy hitter even before Covid. And what if the brain fog or some fraction of it is still there? It's like walking into a fight with one arm tied behind your back. It will still hinder you. And were you even really that competitive beforehand?]

  3. Good luck. #2 is not to discourage you. Is my honest, Bayesian assessment.

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