Teach reasoning, not memorization / pattern matching.
Caveat: You need primitives to reason with — memorizing explicitly helps bootstrap generalization.
School teaches procedures/program templates to memorize.
We are teaching mathematics as if people were LLMs… The same exercise over and over again, without talking about the concepts, intuitions, principles, … → we are not teaching them to think, how to use their intelligence, just to be a tool.
We’re teaching composed programs as if they were primitives.
Reasoning would be deriving the procedure.
We should teach composition, not memorized compositions.
In school you don't learn (let alone learning to learn), you memorize. You guess the teacher's password.
In fact, what school is doing is in many cases actively harmful.
I suspect the most dangerous habit of thought taught in schools is that even if you don’t really understand something, you should parrot it back anyway. One of the most fundamental life skills is realizing when you are confused, and school actively destroys this ability - teaches students that they “understand” when they can successfully answer questions on an exam, which is very very very far from absorbing the knowledge and making it a part of you. Students learn the habit that eating consists of putting food into mouth; the exams can’t test for chewing or swallowing, and so they starve.
Link to original
Much of this problem may come from needing to take three 4-credit courses per quarter, with a textbook chapter plus homework to be done every week - the courses are timed for frantic memorization, it’s not possible to deeply chew over and leisurely digest knowledge in the same period. College students aren’t allowed to be confused; if they started saying, “Wait, do I really understand this? Maybe I’d better spend a few days looking up related papers, or consult another textbook,” they’d fail all the courses they took that quarter. A month later they would understand the material far better and remember it much longer - but one month after finals is too late; it counts for nothing in the lunatic university utility function.
Link to original
It may be dangerous to present people with a giant mass of authoritative knowledge, especially if it is actually true. It may damage their skepticism.
So what could you do? Teach students the history of physics, how each idea was replaced in turn by a new correct one? “Here’s the old idea, here’s the new idea, here’s the experiment - the new idea wins!” Repeat this lesson ten times and what is the habit of thought learned? “New ideas always win; every new idea in physics turns out to be correct.” You still haven’t taught any critical thinking, because you only showed them history as seen with perfect hindsight. You’ve taught them the habit that distinguishing true ideas from false ones is perfectly clear-cut and straightforward, so if a shiny new idea has anything to recommend it, it’s probably true.
Link to original
Maybe it would be possible to teach the history of physics from a historically realistic point of view, without benefit of hindsight: show students the different alternatives that were considered historically plausible, re-enact the historical disagreements and debates.
Link to original
Maybe you could avoid handing students knowledge on a silver platter: show students different versions of physics equations that looked plausible, and ask them to figure out which was the correct one, or invent experiments that would distinguish between alternatives. This wouldn’t be as challenging as needing to notice anomalies without hints and invent alternatives from scratch, but it would be a vast improvement over memorizing a received authority.
Link to original
A relatable 1 annecdote
i remember being punished in school for reading infosec textbooks instead of memorizing the exact dates of various historical events the teacher haphazardly fetishized. i remember being taught we were going to ‘run out of oil’ with 100% confidence and we had to start adapting our lifestyle then and now. i remember a teacher who would physically hurt me because she didn’t like how easily i could get away with skipping over her terribly bespoke and overly verbose methodologies for whatever made up math problems the district’s curriculum entailed. after the misery became unbearable and the color had been fully drained from my eyes i simply refused to do the work and failed the class. this was particularly annoying because administrators had a habit of lying to students by telling them that if they had a poor grade in a class (in middle/high school!), future employers would somehow know about this and would refuse to hire you and you would die alone in a ditch of your own irredeemable digging. luckily i later scored perfect on my math sat which spared me from the apocryphal grade-demons. even so i wasted years before i realized college was fake and was not made for something like me no matter what everyone around me said to the contrary. - near
It’s good to keep the computations manageable and let the ideas come out - Gilbert Strang 1
I think math is easy as soon as you realise that math at school is just hitting numbers into your calculator to use predefined functions - Lukas
“Tedious finger-exercise“