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A post in Professor Doom's blog contains the following exchange:

Pitchman: “What we know for sure is, the students who score high enough for College Algebra usually took courses past College Algebra in high school. The students who don’t quite make it into College Algebra took it in high school, and so on down the line.”

Faculty: “Wait a minute. You mean there’s hard evidence for what we know to be true: that to achieve a goal in learning, you must push past that goal to succeed? Why haven’t these results been published?”

The general idea seems to me to be that if one wants to increase one's comfort with problems at a particular difficulty level, then one should do problems beyond that difficulty level.

Is there a name for this effect? And if so, what are the key findings?

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I found a paper that contains a graph that shows the loss of algebra knowledge over time:

For those who took courses beyond calculus, the drop was neglible over 50 years.
For those who took only calculus, their knowledge dropped from 90% to 75% over 50 years.
For those who didn't take calculus, their knowledge of algebra dropped below 50% after 50 years.

Abstract

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 (e.g., SAT scores, grades) have trivial effects on the rate of performance decline.

Citation

Bahrick, H. P., & Hall, L. K. (1991). Lifetime maintenance of high school mathematics content. Journal of Experimental Psychology: General, 120(1), 20–33. https://doi.org/10.1037/0096-3445.120.1.20

Relevant post by Bryan Caplan: https://www.econlib.org/archives/2012/10/does_high_schoo.html

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