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.
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.
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: