THEN SAID A teacher, Speak to us of Teaching.
And he said:
No man can reveal to you aught but that which already lies half asleep in the dawning of your knowledge.
The teacher who walks in the shadow of the temple, among his followers, gives not of his wisdom but rather of his faith and his lovingness.
If he is indeed wise he does not bid you enter the house of his wisdom, but rather leads you to the threshold of your own mind.
The astronomer may speak to you of his understanding of space, but he cannot give you his understanding.
The musician may sing to you of the rhythm which is in all space, but he cannot give you the ear which arrests the rhythm nor the voice that echoes it.
And he who is versed in the science of numbers can tell of the regions of weight and measure, but he cannot conduct you thither.
For the vision of one man lends not its wings to another man.
And even as each one of you stands alone in God’s knowledge, so must each one of you be alone in his knowledge of God and in his understanding of the earth.
That's called "zone of proximal development" in pedagogy.
However, "that which already lies half asleep in the dawning of your knowledge" is often a consequence of things that were revealed to you previously.
When you learn, you could probably make any specific step alone, but you cannot make all the steps alone.
The nice thing about learning, the more you do it, the easier it becomes.
In germany you have the concept of specialization and you can choose different areas.
The syllabus had all the methods we had to know, and I learned them all through youtube on a channel called NumericalMethodsGuy. I stopped attending class and just went to labs (which were really just matlab assignments in numerical methods) and turned in assignments, and wrote exams. I got an A.
Key takeaways:
* Mathematics is hard. Much harder than most other subjects (except physics which is mostly hard because of all the math involved).
* University is not like primary/secondary school. It is a place where you need to learn how to take responsibility for your own learning. Ideally, you learn how to become an adult.
Every one of my classmates began their degrees from a different place. They excelled in some areas and struggled in others. Many dropped out. This led me to believe that most secondary schools do not fully prepare their students to study mathematics. Having said that, those of us who did make it weren't exactly geniuses. Just people who got used to it.
"Young man, in mathematics you don't understand things. You just get used to them." -- John von Neumann
I've been tutoring high school students in math (and other subjects) for 8 years now. One thing I'd like to add is that I can tell who will succeed at mathematics and who will struggle just by watching them work from across the room.
The students who succeed are the ones who can sit there and focus for hours at a time. The ones who struggle do so because they can't focus for more than five minutes and then start socializing. I think one of the biggest issues for people studying math is that they either can't focus (due to ADHD) or they have math anxiety which fills them with dread any time they try to study. This dread can be so overwhelming that they will do anything they can to avoid it, so they stop studying and do something else.
When I was studying math I was spending upwards of 40 hours per week working on homework. Although this is normal for anyone with a full-time job, it's an unfathomable amount of time to be studying math for those who struggle. This is really what it takes though. An unrelenting drive to figure things out.
It really was the savior of me. I can focus for hours when I get into something, but its the friction of starting up that I found other people really helped with. There are formal names for it, like "body doubling", but I didnt know of that, I just knew it was critical to work together with others to get stuff done, which as the above correctly writes, absolutely must be done.
I learned complex math concepts before the university like math i needed for 3d graphics without issues.
If we as a society think its okay to have such an entry barrier, i disagree and therefore really like having an AI tutor.
Understanding things in math is definitely possible, and whoever says otherwise probably sucks as a teacher. Yes, it is possible to just say a lot of random stuff, and the best students succeed to figure it out anyway. But if you take care to actually explain how it works, the proportion of successful students increases dramatically.
I failed in university at math. Why? Because the tutors had not the time to help me. My level of math was not the same as the other students as i was not in the math part of a gymnasium.
I struggled and wasted a lot of time and energy to even find good explanations.
And when i had a math group, one girl was super nice but knew so muchmore than i did because of her math in gym. Professors asumed so much knowledge and no one cared to try to help people.
Best help were people from india on youtube with bad english.
And the most ridiculous part: Every year around the globe people teach this level of university math to probably millions of students. We should have the perfect free educational platform which teaches everyone perfectly already because so many tutors and professors lecture on the same topics over and over and over again. Our educational system is a joke.