The first rule "do not learn what you do not understand" is nonesense in the context of learning mathematics/physics/hard sciences.
If you understand a subject, then it means that you have learned it effectively.
In my experience with teaching at university, I have found that the wish of students to understand before learning is actually a great barrier to comprehend a subject.
In order to build intuition on a subject, a student should first try to apply it, play with it (without understanding it) and then understanding will come. But I have yet to witness a student which understand a subject without being able to use/apply the subject.
Take the notion of electric field. You could try to understand it before learning how to use it. Good luck. You are not a fish, so you probably have no sensors of electric field on your skin, and thus you have no prior notion to cling to.
I contend that it is virtually impossible. Or you could try to use the concept of electric field to compute forces on charged particle, or compute the electric field created by a charge, or you could build a program that represent the electric field in space. Doing this requires no understanding, just boring substitutions in the definition of the electric field. But doing that forces you to build understanding (its a vector, it changes direction with the sign of the charges, etc, etc)
Another example is understanding how to bike. You could try to understand how balancing on a bike so that it stay upright, understand how moving the handlebar right makes you turn left or, you could just try to bike, and then understand how it works out.
So my advice: don't try to understand. Do, and do again untill you have learned. Not the other way around.
As a kid I cried at times because learning foreign words, lists of topological places, or events in history was so hard! If I don't get the structure or context I am so lost.
At electrical engineering they still tried to teach me in the same way. Giving me assignments with little context. What saved me in the end were textbooks and the internet.
The easiest way for me to understand new concepts are the subsequent generalizations or extremes. Complex numbers? Quaternions. Gradients? Clifford algebra. Sets? Categories. Fourier transform? Wavelets.
Sure there might be people who benefit from solving the puzzles from their teachers, but it assumes that every mind works the same way.
I'm the same way. At university, I had an instructor for Calculus I who seemed content with just having us memorize formulas and identities and "shortcuts", only to regurgitate them on demand. I barely passed with a C-. Then, I had an instructor for Calculus II who was much the same. I made a D+ in Calc II, which, since Calculus I-III are considered essential courses for engineering students, was not passing. My second time through Calc II, I barely passed with a C-.
Then, I get to Calculus III, and my instructor introduces each concept by showing us a complicated-but-ultimately-intuitive formula for something, then deriving the "shortcut" step-by-step. While most of the students griped that it was boring, I found it quite interesting, and for the first time in my university career I felt like I could grasp calculus. I made an A-.
There are other instances of this, as well, but that one was probably the most dramatic. I don't know what it is. I came up with a handful of analogies just now, but none of them seemed particularly satisfying, so I gave up. Perhaps it's just a quirk, I dunno :)
double this. Practicing is useless, if you dont have motivation where you can apply this knowledge. I wish i could go back in time and learn the math in university once again after failing to understand things in stat and ml. just because missing basics things what i "practiced" in the past without understanding it...
I took understanding as meaning motivation. If I was teaching electric fields I wouldn't start with the equations. Perhaps I'd start by demonstrating what an electric field is (e.g. http://practicalphysics.org/electric-fields.html), then perhaps how it's useful, and then open the discussion of what it's properties are. You can then relate the properties back to the demonstrations and applications.
> So my advice: don't try to understand. Do, and do again untill you have learned. Not the other way around.
This is so so true so some level I think our desire to understand everything beforehand is just a lazy strategy. Probably either to learn only what is easy or what is useful. Sense of wonder and discoverablity which makes putting the effort in a joyous activity is gone from us. Our profit mazi
On the contrary Children just keep practising until they learn. I wish I can go back to stage where my 2 years old daughter is.
If you understand a subject, then it means that you have learned it effectively.
In my experience with teaching at university, I have found that the wish of students to understand before learning is actually a great barrier to comprehend a subject.
In order to build intuition on a subject, a student should first try to apply it, play with it (without understanding it) and then understanding will come. But I have yet to witness a student which understand a subject without being able to use/apply the subject.
Take the notion of electric field. You could try to understand it before learning how to use it. Good luck. You are not a fish, so you probably have no sensors of electric field on your skin, and thus you have no prior notion to cling to. I contend that it is virtually impossible. Or you could try to use the concept of electric field to compute forces on charged particle, or compute the electric field created by a charge, or you could build a program that represent the electric field in space. Doing this requires no understanding, just boring substitutions in the definition of the electric field. But doing that forces you to build understanding (its a vector, it changes direction with the sign of the charges, etc, etc)
Another example is understanding how to bike. You could try to understand how balancing on a bike so that it stay upright, understand how moving the handlebar right makes you turn left or, you could just try to bike, and then understand how it works out.
So my advice: don't try to understand. Do, and do again untill you have learned. Not the other way around.