Throughout my life, I used to ask myself, “How do people invent something?”. In my case, I was specifically concerned with Maths, Physics, and Computer Science.
I would ask myself, “I know how to use these formulas but how did someone come up with these formulas?”.
My teachers’ responses were always the same: “Don’t worry you will understand how it works later on. For now, just learn how to use this formula”. Okay, that was simple. I was already doing it. Learning a bunch of formulas without realizing “what” or “who” they were.
I am talking about even simple things like π. We encounter π in almost every calculation which involves circles but I was never taught what π “is”. I was forced to take a stab at questions directly. My learning never began with the concept that π is the ratio between the circumference and the radius of a circle.
Another example is “i”. I learned that “i” is used to differentiate imaginary numbers from real numbers. What I never learned was how “i” just changes the “direction” we are moving in on our graph. Once it clicks, Euler’s theorem e^(iπ) = -1 starts making more sense. Circular motion makes more sense and waves make more sense.
What we are taught are the tools but not the theory behind the tools. If we don’t learn how the basic tools are made then how can we expect someone to improve or invent new tools in any meaningful way?
Everything in Maths, Physics and Computer Science is connected. All the formulas and theorems are derived from the basic concepts we already know and are familiar with.
I used to believe that this is exactly how I am supposed to be taught. I am just not smart enough to figure out the relationship between all of this stuff myself.
I was proved wrong by the book “Surely You’re Joking, Mr. Feynman!”.
Let me relate one incident where Feynman was invited to give a bunch of lectures in Brazil. During the course of his stay, he learned that the students knew all the formulas by heart but they couldn’t “see” the formulas in real life. He asked them about light polarization in a lake and asked a related question. They were blank. He asked them about polarization in general, they knew every theorem.
What the students failed to do was to observe that lake itself was acting as a mirror. They were unable to make the connection and solve the question which Feynman gave them! Feynman was talking to students who were like me. They knew the theorems but not the intuitiveness and applicability of those theorems! They simply had to apply the theorems they knew by heart in a different setting and they were unable to do it.
This is just one of the interactions of Feynman with students in Brazil. There are a whole lot more in the book.
At the end of his stay, the students asked Feynman to say a few words regarding his stay in Brazil. Feynman agreed to do it only if he was free to say whatever he wanted. He got the approval and hence a big event was organized before he left Brazil. All the famous politicians, educationalists, and students were in attendance.
Feynman got up on stage and said: “No science is being taught in Brazil”.
He backs up his claim by his experience in interacting with the students. He gave specific and convincing examples on why he thinks that is the case. I would highly recommend everyone to give his book a read and specifically focus on the “O Americano, Outra Vez!” chapter.
What is my aim?
I might get a lot of criticism by making it sound as if my professors did not teach me anything. That is not the case! They are all learned and amazing people who taught me everything I know. I just want them to focus more on the intuitiveness of the questions than the theorems themselves. Make the student appreciate the theorem before forcing him to learn the definitions. If the students start appreciating how everything is connected, learning comes naturally and things become more interesting.
What am I planning on doing now?
I am going to try and learn everything “intuitively” first, rather than cramming a bunch of algorithms and ideas in my mind. I will try to make sure I “understand” something before I apply it in practice. It is a very hard task, especially when your whole life you have been used to just learning formulas and applying them.
If you want to look at maths from an intuitive perspective then read the articles on Better Explained by Khalid Azad and watch the videos by 3Blue1Brown on YouTube.
Is the future challenging? Yes. Am I looking forward to it? Absolutely!
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