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While performing mathematically relevant directed actions facilitated key mathematical insights and intuitions for two tasks (Triangle and Gear), directed actions on their own did not lead to superior informal proofs compared to irrelevant actions.
Performing mathematically relevant directed actions (without pedagogical language) will predict student performance on mathematical insights.
Professor Scarf's mathematical insights and formidable mental dexterity were invoked in more than one Nobel citation.
Nash's two embedding theorems are considered classics, providing some of the deepest mathematical insights of the last century.
On its own terms, though, the movie succeeds in dramatizing Mr. Nash's mental illness and, to a lesser extent, his mathematical insights.
The challenge has been to combine its mathematical insights with the kinds of adjustments that are needed to make its models fit the economy's irreducibly human element.
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Learning About Mathematics: What is the most efficient way to gain mathematical insight?
Poincaré credited his sudden mathematical insight to "unconscious work," an ability to mull over the mathematics while he was preoccupied with unrelated activities, like talking to a friend on the bus.
It built mathematical insight!
He felt at any moment he could have the next mathematical insight.
Perhaps most important, both men provided deep mathematical insight into the nature of bluffing when they wrote "Theory of Games and Economic Behavior" in 1944.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com