A Mathematician's Lament [pdf] | Hacker News Reader

A Mathematician's Lament [pdf]

worrydream.com

Any time someone who does not understand the practical outworkings of a given topic tries to "teach" it, you end up with a "presentation" - mostly in the form of propositional statements (dogs are mammals; barges are boats), or key-value pairs (in 1492, Columbus sailed the ocean blue; Caracas is the capital of Venezuela)

Propositional statements and key-value pairs are absolutely vital aspects of learning and understanding - but seeing how those ideas, facts, and statements will work themselves out in practical application is absolutely mandatory for proper understanding of the topic

If you never learn how you might use these facts, you are going to have an incredibly difficult time "learning" them

This is the fundamental problem of why "teaching to the test" (or, for that matter, relying too heavily on multiple-choice, true|false, matching, etc type "tests" to evaluate "learning" or "knowledge") is problematic: I can teach pretty much anyone how to pass a multiple-choice test with zero knowledge of the subject material (did it myself years ago when I passed the first-tier amateur radio licensing exam without ever looking at any study guide)

When you "teach to the test" (which is what, fundamentally, exclusive reliance on propositional statements (and drills over them) and key-value pairs are), you create people who may (or may not) end up being good at trivia challenges ... but have no mental framework for connecting all those dots into anything coherent - iow, you create human-based data lakes: it is all sitting there, but no connective lines have been drawn

To get someone to want to learn, you have to show they why it "matters" - you have to get them to develop intrinsic motivation to learn (vs the purely extrinsic "you have to to pass")

To develop that intrinsic motivation, you need to show the 'why' and the 'where' of the 'what'

And to do that you have to understand it yourself

This isn't exactly what you are saying, but I have noticed that students basically 'overfit' the training data on multiple-choice tests.

A nice use of LLMs might be to rewrite these questions on the fly so they test the same understanding while making it difficult to memorize answers.

Failure to learn seems almost synonymous with overfitting. It’s the failure to distinguish noise from signal due to lack of data (or just lack of imagination and ability to quickly see the general pattern, or even that there might be one).

That would indeed be a nice use of LLMs, although for mathematics I wouldn’t feel I could trust them not to warp the meaning badly. You’d probably have to check each generation carefully by hand.

This ‘key-value’ thing is so true in my experience — speaking with anyone who has no mathematics education beyond the mandatory years, it’s clear that almost everyone thinks of it being a subject of classification and naming rather than understanding and exploring. The standard example of ‘the sort of thing mathematicians do’ probably begins with a recital of the quadratic formula or Pythagoras’ theorem (invariably stated simply as ‘a^2 + b^2 = c^2’, as if each of these symbols has specific mathematical meaning on their own), or perhaps a statement about which triangles are considered ‘isosceles’ or ‘scalene’.

The most common view of mathematicians seems to be that of a slightly more general stamp collector who likes collecting and giving needlessly complicated-sounding names to things.

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