I do recognise that spoilers have an effect ina learning experience but I also have experiences where knowing about a thing hasn't be an obstacle to gaining insights. One of the patterns is knowing what words mean without being too aware of their etymologies. Like hydrogen as "water-maker" or oxygen as "black-maker". One is likely to be aware of charred ruins and the gas that people use to breath but making the connection is separate from knowing about them.
Education is likely to be optimised to be informed about things but the priority of understanding things could warrant for different approaches and fruits.
Also, it's worth noting that in many of these religions "enlightenment" seems to be something like "a feeling of supreme insight without any associated insight".
I vaguely remember reading it in a book (probably by Oliver Sacks) that there are separate parts of brain responsible for "insight" and "feeling of insight", and you can have the latter without the former by stimulating the corresponding part of brain.
So I wonder if teaching mathematics should take a page out of the mystic handbook and focus on giving people exercises and letting them try to figure useful properties and theorems out by themselves. Or, at most, teaching certain theorems only when they are needed, i.e. once a student hits their head enough against an exercise so that the answer to their problems, so simple and clear, seem revealing, inspires awe, insight.
This idea is known as "constructivism". There are both good and bad ways how to do it.
The bad way takes it to the extreme, and tries to make students reinvent millenia of human progress independently. This predictably fails; sometimes such "education" produces teenagers who just recently grasped the mystery of addition.
The good way combines the independent discovery with manipulation, like you described it. The students discover the rules, but the teacher gives them the right exercises that make the discovery almost inevitable. You can speed up the process by having the students discuss their insights with each other (as a side effect, they also learn how to explain, and how to find mistakes in explanations), so when enough students get the idea, the rest of the class will follow.
In practice, the biggest problem is to find teachers who are competent enough to provide all the nudging without giving away the solution. Also, if a student finds an answer that is wrong, but the classmates do not object, the teacher needs to nudge them towards finding the mistake. The wrong answers are difficult to predict, so the teachers need to improvise. That requires solid math skills, and... the truth is that most math teachers, at least in elementary school, actually suck at math.
You would kind of need a whole generation of students educated this way in order to get enough teachers capable of teaching this way. A chicken-and-egg problem. It is not enough to only teach some students this way, because they may choose professions other than teaching... and, ironically, superior math skills would only make that easier. But people are trying; there is a math reform in Czechia based on these principles:
The trouble with acquiring experience is that experience cannot be transmitted. It can only be gained. There is only one way for a child to acquire experience in mathematics – by solving a problem. Any effort to make a pupil’s path to understanding shorter and to try to “pass on the experience” only addresses a momentary situation. No matter how noble our intentions, in reality we are doing the pupil a disservice. The knowledge we pass on to them is formal, and is only stored in their mind temporarily. In effect, it is not knowledge in the true sense of the word.
"a feeling of supreme insight without any associated insight"... I call this a "content-free Aha moment".
Regarding math education, you might look into the Moore Method of teaching topology.
If all I have to do to /reach enlightenment/attain nirvana/understand the nature of God/ is to read a few sentences, why not readily give them to everyone?
Well, technically because it'd be meaningless. The typical analogy is with telling a born-blind person that "seeing the color red is like feeling warmth, and seeing the color blue is like feeling cold". They grasp the words, they get the analogy, they build some very, very fuzzy mental model of what you might be talking about, but they definitely didn't see anything red or blue (or anything at all) by having merely heard the sentence.
The assumption, then, is that at least some people (not all) have the possibility of developing, through careful, directed practice, the ability to perceive things beyond the ordinary senses. Not out of nowhere, but because the potential is already there, it just needs work. Once they do, they'll also notice the "something" that was given that weird name, "red", by those who noticed it before. And now that they too have the "knowledge" of what "red" is, they can begin talking with other "red-perceivers" about mindbogglingly esoteric unintelligible things such as "shades of red", "pigments", "the color wheel", "the RGB standard", "visible colors that don't exist in the electromagnetic spectrum" etc., whatever those might be, and refining them further.
So, it isn't that you cannot have access to the sentences. In fact, you can find plenty of very detailed descriptions of what "Emptiness", "Nirvana", "Buddha Nature" and the like are. But they are all, for those without the actual "knowledge" (perception) of them, just so many variations of "red is warm".
The mind of the enlightenment era mathematician might have been the ultimate tool ever devised for creating pointless and convoluted connections between pi and everything else.
i - Secret based religion
There are certain Buddhist traditions, e.g. Dzogchen, in which "enlightenment" or some other desired state or status is predicated upon knowing some hidden knowledge. This is most popular in Buddhist faiths but by no means confined to them, some gnostic traditions are also fund of secret knowledge.
This begs a question for the would-be believer:
If all I have to do to /reach enlightenment/attain nirvana/understand the nature of God/ is to read a few sentences, why not readily give them to everyone?
The answer to this has something to do with "mind preparedness", one is not ready to understand until they have some prerequisite baggage. But still, why not give them to everyone first, then tell them to go get the prerequisite knowledge. After all, the prerequisites might be different for everyone, this way, as soon as they have them, things will instantly click, and they won't have to /meditate/chant/pray/ for longer than necessary.
To which the crazier believers answer something like:
But the saner ones say something like:
Sound like a load of rubbish? Ok, I agree, but the methodology these sects invented for causing a feeling of revelation might be quite generic and ingenious. Let me give a brief summary of how this goes:
Also, it's worth noting that in many of these religions "enlightenment" seems to be something like "a feeling of supreme insight without any associated insight". Granted, the religious people would say "there is associated insight, we just can't explain it in words, or actions, to anyone but those that are enlightened", but I've taken large enough doses of acid not to fall for that one, there are ways to generate "a feeling of supreme insight" while revealing nothing of value about the world.
So the above, I would claim, is a tried and tested pattern for generating something that feels like awe or understanding or enlightenment or insight or something equally blissful and important.
ii - High school math
Did you ever ponder some mathematic formula or problem and had a moment when things click and felt "Ohhh, ****, that makes total sense" and your worldview slightly but notably shifted forever. Maybe something around some properties of a circle or of
e. Maybe something around edge-cases for probabilities (sailor problem, sleeping beauty problem, Monty hall problem are common).I had this feeling a bunch of time, the one I most vividly remember is when I figured out how to compute the probability of an outcome happening once, for multiple independent events that have the same range of outcomes (e.g. what's the change of rolling 6 at least once if you roll 6 6-sided dices). Usually, it's something simplistic, because most of us aren't really able to "get" complex math on an intuitive enough level to deduce things enough inferential steps away for them to be insightful.
Most of these things we already "know", we are told about them in school, usually much earlier than we could figure them out on our own. The only reason we can "relearn" them is, I assume, that we forget them due to how horribly they are "taught" to us.
So I wonder if teaching mathematics should take a page out of the mystic handbook and focus on giving people exercises and letting them try to figure useful properties and theorems out by themselves. Or, at most, teaching certain theorems only when they are needed, i.e. once a student hits their head enough against an exercise so that the answer to their problems, so simple and clear, seem revealing, inspires awe, insight.
Mind you, I'm not sure this would be "good" in absolute terms, or good for the current (lacklustre) value structure of society, but I think it would make more people love math a lot more. Two anecdotes lead to the solidification of this idea as worth investigating:
1. I and many other people I know that are decent at math, but not exceptionally good at it, seemed to have really loved math up until somewhere around the 3rd to 7th grade. Incidentally, this seems to be the place where math "picks up speed" and it's almost impossible to stay "ahead of the teacher" and discover stuff for yourself. It's easy to reason your way to insights when learning Euclidean math, there's a reason Greek rationalists thought everyone "knew" it inherently. Not only is Euclidean math simple, you've got 6 years to learn it, and you don't even cover most of it.
It's much more difficult to even start fathoming the need for calculus, let alone "discover" stuff about it on your own, yet we are thrust into it right after Euclidean math in a much more rapid way, with 1000+ years of in-between mathematics glanced over. So losing interest in math at the start of or soon before high-school might be the fact that we go from a bunch of moments when "things click", when we feel insight, to being told facts we have no way of appreciating or "getting" at a level where they seem important.
... but maybe it's testosterone, that could be it too, it also checks out in terms of timing.
2. Being very smart (at least as far as an IQ test can measure) is heavily correlated with being a good mathematician and, seemingly, with loving math. The extent of this is surprising since other fields of thought seem less reliant on intelligence and much more reliant on luck and "hard work". Even closely related fields like computer science. Even more so, "loving" any given field doesn't seem predicated on how smart people are, except for mathematics, where the people that actually "love" math are usually smart to the point of associated mental illness.
The "love" bit is confounded by people social signalling that "they love math" a lot more than they social signal love for other sciences. But I'd protest that this is an argument for the "very smart people love math" hypothesis since math is essentially useless in most jobs and areas of life, but intelligence isn't, so social signal for intelligence are very useful. Thus I'd propose that people try signalling "I am smart" by saying "I love math" is actually the result of a shared observation of a reality (very smart people usually love math and only very smart people can love math).
Granted, this is all heavily anecdotal. But if true, one explanation for it would be that very smart people are able to "stay ahead of the teacher" when it comes to math all the way through school and high-school, and thus get exponentially more "eureka" moments than the just-slightly-dumber people that are almost always one step behind the teacher.
... or maybe it's just that, like many other things, mathematical intelligence has exponential returns. Von Neuman gets to invent most of what we know as "anything of scientific relevance since the 40s" and the next 100,000 smartest people get to invent "most of the scraps that were left to them by Von Neuman"... maybe a slight exaggeration, but you get the point, there's certainly a case to be made for exponential returns on intelligence. I strongly protest against that case, but that's an argument for another day.
Still, I would very much like to know if this style of "letting kids figure it out for themselves" has been tried with mathematic education and what the results are, once controlled for confounders.