How to Learn Math
and How to Teach it
For people who enjoy watching videos, the lesson is embedded on the right from youtube.
For people who enjoy reading, a more detailed article is available below. 

Introduction to How to Learn Math
So we figured out what math is (kind of: what is math)… And we are convinced that we want to learn it, either for:
1. Its own sake and beauty, or
2. For improving science and engineering through accurate and precise analysis using mathematics as a language.
1. Its own sake and beauty, or
2. For improving science and engineering through accurate and precise analysis using mathematics as a language.
What You Need to Know to Understand How to Learn Math
You need to have a decent idea of what math is. In summary math is the study of necessary synthetic truths through rational deductive reasoning on abstract ideal entities. If that sounds like gibberish, watch the first video titled “what is math” or read the note package (click here).
How to Learn Math: Core Lesson
Georg Cantor: “In mathematics the art of proposing a question must be held of higher value than solving it.”
So now you are taking a course and you want to know how to learn calculus or any other math topic. You are wondering what is the best way to approach it? Well: Since math is dependent on rational deductive reasoning, we know that you MUST be capable of demonstrating (1) rational deductive reasoning to prove any important theorem you are learning, and since the entities of math are abstract ideal entities, you must be capable of (2) defining exactly what concepts you are dealing with.
For example: Let us imagine you were learning calculus…you will therefore need to know the definition of a derivative, A limit, An integral, A function, Etc… and you will need to know the deductive reasoning behind the Fundamental theorem of calculus, the different rules to differentiate, the different rules to integrate, the mean value theorem, etc… Thus your understanding will be verifiable if you are capable of demonstrating knowledge of definitions of entities and deductive logic of theorems.
Now this is only an alright answer, it gives us an indication if we learned something or not, BUT: this still doesn’t tell us exactly “HOW to learn math”! it tells us how we will be after learning math!
So now the question is: what is the best way to learn math definitions and mathematical deductive logic? Well, I will not discuss the different methods to do it, I will simply and plainly state the best way to do it:
The best way to learn a definition of an entity is to define it yourself, thus through your own critical thinking you come up with a definition for an intuitive concept you had. And the best way to learn logical thinking is to go through the process of connecting premises with conclusions, or in other words, is to do the logical thinking yourself.
You might be annoyed from my answer, and I am slightly embarrassed to have to explain this, but the best way to learn thinking is to think! And the best way to learn mathematics is to behave like a mathematician, and the behavior of mathematicians is all about critical defining of concepts and critical deductive reasoning of theorems.
Thus the best way to learn mathematics is to go through a process that helps you give good definitions of concepts and (2) gives you an ability to think through the logic behind different theorems and methods. This process must give YOU the responsibility of finding the definitions and theorems.
All teachers I have experienced were in the habit of TELLING definitions and theorems to students. Being told definitions and theorems DOES NOT give you the ability to think them through, it gives you the option to memorize them. Which means you are an advanced form of a parrot. Nothing more.
For example: Let us imagine you were learning calculus…you will therefore need to know the definition of a derivative, A limit, An integral, A function, Etc… and you will need to know the deductive reasoning behind the Fundamental theorem of calculus, the different rules to differentiate, the different rules to integrate, the mean value theorem, etc… Thus your understanding will be verifiable if you are capable of demonstrating knowledge of definitions of entities and deductive logic of theorems.
Now this is only an alright answer, it gives us an indication if we learned something or not, BUT: this still doesn’t tell us exactly “HOW to learn math”! it tells us how we will be after learning math!
So now the question is: what is the best way to learn math definitions and mathematical deductive logic? Well, I will not discuss the different methods to do it, I will simply and plainly state the best way to do it:
The best way to learn a definition of an entity is to define it yourself, thus through your own critical thinking you come up with a definition for an intuitive concept you had. And the best way to learn logical thinking is to go through the process of connecting premises with conclusions, or in other words, is to do the logical thinking yourself.
You might be annoyed from my answer, and I am slightly embarrassed to have to explain this, but the best way to learn thinking is to think! And the best way to learn mathematics is to behave like a mathematician, and the behavior of mathematicians is all about critical defining of concepts and critical deductive reasoning of theorems.
Thus the best way to learn mathematics is to go through a process that helps you give good definitions of concepts and (2) gives you an ability to think through the logic behind different theorems and methods. This process must give YOU the responsibility of finding the definitions and theorems.
All teachers I have experienced were in the habit of TELLING definitions and theorems to students. Being told definitions and theorems DOES NOT give you the ability to think them through, it gives you the option to memorize them. Which means you are an advanced form of a parrot. Nothing more.
Dilemma in Teaching Independent Learning and Thinking
This answer puts us in a weird position, we want to teach a student to “think for himself” but then we are really teaching him how to think at the same time! So is he really thinking for himself? Socrates was put in this same dilemma. It appears to be somewhat solvable in one way: what we can do is set up the right environment for “thinking for yourself” and hoping that the student will actually put in the effort to think! Now our goal is still to teach them math, and math is a branch of knowledge related to abstract ideal entities. So the right “environment” for it is really: in the minds. The best way to get someone thinking and using his mind without TELLING him what to think, is to ask him good questions. The questions have to be vague enough to allow room for free thinking and specific enough to force the student into a direction picked by the teacher (ie: toward understanding the fundamental theorem of calculus). This technique of educating people is called the Socratic Method. Famously used by Socrates to educate the youth, it was so good that he was executed for “corrupting the youth” (ie: educating them). Hahah. Socrates is the man who taught Plato, which in turn taught Aristotle and Zeno. The three of which formed 3 out of the 4 most important classical schools of thoughts (Platonic, Peripatetic, and Stoic schools. The fourth is the Epicurean school.)
Summary of How to Learn and Teach Math
(1) You need/want to learn math
(2) Math is the study of necessary synthetic truths through rational deductive reasoning on abstract ideal entities
(3) You therefore need to understand:
a. Definitions of abstract ideal entities
b. Rational deductive reasoning
(4) Best way to understand these (a & b) is through your own critical thinking
(5) Best way to induce guided critical thinking is through asking questions (ie: teacher asks relevant questions, student answers questions)
(2) Math is the study of necessary synthetic truths through rational deductive reasoning on abstract ideal entities
(3) You therefore need to understand:
a. Definitions of abstract ideal entities
b. Rational deductive reasoning
(4) Best way to understand these (a & b) is through your own critical thinking
(5) Best way to induce guided critical thinking is through asking questions (ie: teacher asks relevant questions, student answers questions)