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Comment by Radix | Hacker News Reader

I think you make an important point, but I'm not sure the post takes any particular focus on visual representation other than a graph is generally an easier way to intuit what a derivative is. His repeated use of the word concept suggests insight for him requires a more general abstraction.

As an aside, why do or did people claim there is visual learning aside from spatial learning? I don't experience visual and spatial imagination as different things. (With reasoning about time always assumed.)

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[...] other than a graph is generally an easier way to intuit what a derivative is.

Even that is a matter of personal preference. I honestly believe it's easier to get the concept of a derivative by linking it to instantenous velocity.

There are many different ways of thinking about mathematical concepts like derivatives. The more you know, the more deeply you know them, the better.

Here's a random example: Marsden and Weinstein define derivatives in their out-of-print textbook Calculus Unlimited without limits. The tangent to a graph at the point x is the boundary between two line pencils, one of lines entering the epigraph at x, the other of lines leaving. There's no limit-taking of chords. It's a simple and neat definition that connects with classical notions of tangency.

In his essay On Proof and Progress in Mathematics, Thurston lists a dozen other definitions or conceptions of derivatives in his personal arsenal, some very sophisticated. But even those among his definitions that are elementary and have roughly the same scope there is a difference in their psychological affordances, and that can make all the difference.

why do or did people claim there is visual learning aside from spatial learning?

I'm pretty sure that those are distinct neurological processes, as revealed by the differing individual deficits that patients can have after suffering strokes. But I don't have the medical references at hand, and you have certainly seen many sources that combine writing about both, as I have.

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