no ideas but in things

[ajschumacher]

title from William Carlos Williams' poem(s): https://allpoetry.com/A-Sort-Of-A-Song

"I think that it is a relatively good approximation to truth - which is much too complicated to allow anything but approximations-that mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure. But, once they are so conceived, the subject begins to live a peculiar life of its own and is better compared to a creative one, governed by almost entirely aesthetical motivations, than to anything else and, in particular, to an empirical science." (John von Neumann, "The Mathematician" part two, https://mathshistory.st-andrews.ac.uk/Extras/Von_Neumann_Part_2/)

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Lockhart in The Mathematician's Lament:

"Mental acuity of any kind comes from solving problems yourself, not from being told how to solve them." (pages 53-54)

"Mathematics is about problems, and problems must be made the focus of a student’s mathematical life." (pages 60-61)

[ajschumacher]

"Of course it's no better (in fact, it's substantially worse) to pass along a population of students who've developed some wispy sense of mathematical meaning but can't work examples swiftly and correctly. A math teacher's least favorite thing to hear from a student is “I get the concept, but I couldn't do the problems.” Though the student doesn't know it, this is shorthand for “I don't get the concept.” The ideas of mathematics can sound abstract, but they make sense only in reference to concrete computations. William Carlos Williams put it crisply: no ideas but in things." (page 58, Ellenberg)

[ajschumacher]

https://www.edutopia.org/article/it-time-drop-finding-main-idea-and-teach-reading-new-way

[ajschumacher]

Risk of teaching “thinking skills” as becoming content-free...

[ajschumacher]

knowledge

in Game Changers, page 96, examples: naming something Netflix and Chill without knowing what it means, the Chevy Nova (no va) in Mexico

[ajschumacher]

and E. D. Hirsch, Core Knowledge

[ajschumacher]

"If I had to describe my conclusion [about how to study] in one word, I'd say examples. They are, to me, of paramount importance. Every time I learn a new concept (free group, or pseudodifferential operator, or paracompact space), I look for examples—and of course, non-examples. The examples should include, whenever possible, the typical ones and the extreme degenerate ones." (pages 61-62, Halmos, I want to be a mathematician)

[ajschumacher]

"But let's get back to teaching by challenging. An intrinsic aspect of the method at all levels, elementary or advanced, is to concentrate attention on the definite, the concrete, the specific. Once a student understands, really and truly understands, why 3x5 is the same as 5x3, then he quickly gets the automatic and obvious but nevertheless exciting unshakable generalized conviction that "it goes the same way" for all other numbers. We all have an innate ability to generalize; the teacher's function is to call attention to a concrete special case that hides (and, we hope, ultimately reveals) the germ of the conceptual difficulty." (page 272, Halmos, I want to be a mathematician)

[ajschumacher]

"For Dieudonné the important result is, I think, the powerful general theorem, from which it is easy to infer all the special cases you want; for me the greatest kind of step forward is the illuminating central example from which it is easy to get insight into all the surrounding sweeping generalities." (page 325, Halmos, I want to be a mathematician)

[ajschumacher]

"29. "Conception cannot precede execution." (Maurice Merleau-Ponty, Sense and Non-Sense) Art is a process of discovery through making, and our ability to discover is generally greater than our ability to invent. Think of your work process as a form of travel. Look for the things you don't know, the things that are revealed or inadvertently uncovered. It is easier to find a world than to make one."

https://planspace.org/20210131-101_things_to_learn_in_art_school_by_white/

[ajschumacher]

John D. Cook says it's easier to go from specific to general than from general to specific: https://www.johndcook.com/blog/2024/05/04/abstraction-ladder/