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There are many "rules" for speed arithmetic.

List of some reference links showing speed methods or rules:

Many of these are clever, some are insightful, some are limited.

Are there any authoritative guides that show which types of speed arithmetic methods are actually valuable for students in higher mathematics?

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    $\begingroup$ If you have any connection with any of these sites, please state it. $\endgroup$
    – Sue VanHattum
    Nov 13, 2023 at 2:04
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    $\begingroup$ I have no affiliation or connection that I know of. My goal in finding credible resources for this task is also independent of the sites. I’m a (math) dad of 4 girls, one is freshman, one middle school, one kindergarten age, and a 4 yr old. $\endgroup$ Nov 13, 2023 at 4:18
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    $\begingroup$ I would expect that speed at doing arithmetic calculations is completely irrelevant for "higher mathematics". "Higher mathematics" (whatever it means) has almost nothing to do with making arithmetic calculations in the sense of OP. $\endgroup$
    – Dan Fox
    Nov 13, 2023 at 9:53

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I agree with @DanFox's comment, that speed in arithmetic has no bearing on how well someone can learn higher mathematics. And yet, there are some methods for mental math that help one see relationships.

Beast Academy, an excellent curriculum for elementary school level math, teaches students to use methods that work best when you visualize. That visualizing is helpful with higher mathematics. [Until a few weeks ago, I had no connection with them. Now I am in the middle of applying to teach a course for Art of Problem Solving, the company that produced this curriculum.]

For example, how would you find 652 in your head? I imagine a square, 65 by 65. Now I take a strip, 5 by 65, and move it from the top to the right side (rotated to lie alongside the big rectangle). If I take off a 5 by 5 square at the top of that strip, I now have a rectangle that is 60 by 70. So 652 equals 60*70 +25, which is 4225.

Doing mental math where you understand why each step works will help with algebraic reasoning.

So you would want to evaluate each method. If it feels like a trick, it might be a fun way to do the arithmetic, but probably is of no use further along. It it feels like a method that makes sense, perhaps it will be helpful long-term.

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