How can students truly understand a mathematical formula through first principles, rather than just rote memorization? For example, can the Pythagorean theorem be reduced to the most fundamental facts of 'right triangles and area'?

This is an excellent question, and it's precisely the key to learning mathematics, or any science for that matter. Many people think mathematics is a game for geniuses, but in large part, it's just a matter of method. Rote memorization treats math as a "manual" to be recited, while true understanding treats it as a "Lego world" you can build yourself.

Let's take the Pythagorean theorem (a² + b² = c²) you mentioned as an example, and see how we can "play" with it instead of just "memorizing" it.

Step One: Back to the Most Basic Scenario

Forget the formula. We now only have two fundamental "common sense" facts that require no proof:

We know what a right-angled triangle is (it has one 90-degree angle).
We know how to calculate the area of a square (side × side).

Alright, let's begin.

Step Two: Hands-on "Puzzling"

Imagine you have four identical right-angled triangles. Their shorter sides are a and b respectively, and the longest side, the hypotenuse, is c.

Now, let's play a puzzle game. Find a large board and arrange these to form a large square. How? Connect the a side of one triangle to the b side of another, so they point outwards like a windmill.

You see, after assembling them, we get a large square composed of four triangles and a hollow space in the middle.

(You can visualize this, or draw it on paper)

Step Three: Calculate "Area" from Two Different Angles

Now comes the crucial step. We'll calculate the total area of this "large square" using two methods.

Method One: The Most Direct Calculation What's the side length of this large square? From the diagram, you can see its side length is exactly one a side plus one b side. So the side length is (a+b). Therefore, the total area of this large square is: (a+b) × (a+b), which is a² + 2ab + b². That's simple, right? It's just the expansion of multiplication learned in junior high.

Method Two: Calculate by Breaking It Down What is this large square composed of? It's made up of the "four triangles" we just placed and the "hole in the middle."

The area of one triangle is: (base × height) / 2, which is (a × b) / 2.
The area of four triangles is: 4 × (ab/2), which is 2ab.
What shape is the hole in the middle? It's also a square! Its four sides are exactly the hypotenuses c of the four triangles. So, the area of this small square in the middle is: c × c, which is c².

So, calculated this way, the total area of the large square is the sum of the areas of these pieces: 2ab + c².

Step Four: The Moment of Truth

We just calculated the area of the same large square using two methods. Since it's the same object, the areas must be equal! Therefore:

a² + 2ab + b² (Result from Method One) = 2ab + c² (Result from Method Two)

Now, let's remove 2ab from both sides of the equation. What's left?

a² + b² = c²

You see, the Pythagorean theorem emerges just like that. It's not a "spell" you need to memorize; it's a conclusion naturally derived from the more fundamental fact of "calculating geometric areas." As long as you accept that the area of a square is its side squared, and that the whole is equal to the sum of its parts, you will inevitably arrive at the Pythagorean theorem.

How to Apply This Thinking to Other Formulas?

This process of "reduction" is the essence of first principles thinking. Whenever you encounter another formula in the future, you can try to "attack" it this way:

Ask yourself: What is this formula for? What relationships does it describe? (e.g., the Pythagorean theorem describes the relationship between the three sides of a right-angled triangle)
Then ask: What are the most basic elements that make up this formula? (For the Pythagorean theorem, it's "side lengths" and "right angles")
Try to re-derive it using simpler knowledge you already possess. Can you "construct" it using diagrams, decomposition, combination, or a simple thought experiment? (Just like the puzzle game we just played)

For example, when you learn the formula for the area of a circle, πr², you can imagine cutting a circle into countless thin "pizza slices," then arranging these slices alternately. It will increasingly resemble a rectangle. The length of this rectangle is half the circumference of the circle (πr), and its width is the radius (r), so the area is πr².

This process might feel a bit slow, even "clumsy" at first, but once you get used to it, your understanding of knowledge will undergo a qualitative change. You will no longer be a "user" of knowledge, but a "creator" of it. This will not only help you truly understand mathematics but also cultivate a core ability for deep thinking and solving unknown problems.