> You still need spaced repetition for actual retention of the skill. Though it is easier to recover a forgotten well learned skill.

This isn’t true. People forget the most advanced skills they learned unless they use them regularly but they don’t forget the ones that are necessary for that last skill. If you ever knew calculus you’ll remember algebra after decades without using it. I haven’t spoken German in most of a decade but I can still read it fine. Spaced repetition is the most efficient way to durably learn something but it’s not necessary.

Do you have any sources for your statement? Especially context switching. Where it is terrible at very short notices (because cohesion is low), it is beneficial for introducing spacing while running maximum allowable workload.

People tend to forget unpracticed skills, not oldest, and most complex parts (least compressible, biggest) first, not necessarily hardest. And the forgetting is exponential, depends on how well the material is presented (cohesion specifically).

Exponentials flatten a whole lot.

You do forget a little still over time. Ask a 40 year old who knew algebra and calculus really well and does not use it often if at all. (I do sometimes, so I don't count.)

Stability and accuracy are also separate variables.

Algebra is not one thing, it's thousands of memory chunks. As a probe, think if you remember Fundamental Theorem of Algebra which is a keystone, but slightly tricky. Compare to whether you can solve ordinary differential equations of second kind, and whether you can solve an equation involving logarithms of rational non-negative numbers. (I picked random not absolute basics from high school, 101 and 202. Bonus points if you spot the stinker.)

--

I always thought of even latest versions of SuperMemo like Antikythera mechanism for learning. It's a great model even if principle is not yet properly researched.

Lifetime maintenance of high school mathematics content.

Harry P Bahrick, Lynda K Hall Journal of experimental psychology: general 120 (1), 20, 1991 An analysis of life span memory identifies those variables that affect losses in recall and recognition of the content of high school algebra and geometry courses. Even in the absence of further rehearsal activities, individuals who take college-level mathematics courses at or above the level of calculus have minimal losses of high school algebra for half a century. Individuals who performed equally well in the high school course but took no college math courses reduce performance to near-chance levels during the same period. In contrast, the best predictors of test performance (eg, Scholastic Aptitude Test scores, grades) have trivial effects on the rate of performance decline. Pedagogical implications for life span maintenance of knowledge are derived and discussed.

Correct, later courses force you to practice earlier skills. This is the mechanism these researchers stumbled upon. Plain spaced repetition explains a lot. SAT is definitely not meant to test performance of learning, though some people with very high SAT scores may be a bit more efficient or know tricks that are not widely taught - SAT is too easy to differentiate them.

Related knowledge strengthens already known things, and once critical maximum stability/retrievability is reached (optimally 7 or so precisely spaced rehearsals) it is pretty much cemented. Without optimal spacing, probably quite a few more repeats. At maximum stability/retrievability the exponential maintains "flat" form for a very long time. A refresher may be needed to get facile again, otherwise it may take a short while to remember and there may be mistakes. Even a trivial refresher will work though, chances to use basic algebra are many.

Principles SuperMemo SM-17 algorithm puts in quantification. Here's a link with all the history and references to some other research: https://www.supermemo.com/en/articles/history

Thanks for the paper, though it's slightly wish washy, but still useful.

I only learned math up to maybe the level of a decent BA student but I feel like when you take a math course you learn how to “do a trick” (pass the exam) and then actually learn the material when you take the next course, where you need to figure out which parts of your last class you apply to “do the trick” in the current class.

So my question is what level of calculus is retained, and if students who take calculus then advanced calculus or introductory analysis or whatever you’d call it retain more.

Spaced repetition does not necessitate spending few hours learning per day. That’s not what the research says at all.

Even one to two hours is enough of an optimal schedule is used. It takes longer time if the material is badly presented or not practiced.