In mathematics learning, how can one truly understand formulas through first principles?

Hey friend, that's an excellent question! It shows you're no longer content with just 'rote memorization,' which is a crucial step in mastering any challenging subject. Let me share my experience; I hope it helps you.

Forget the intimidating term 'first principles.' Instead, think of it as 'getting to the bottom of things' or 'returning to the origin.' In mathematics, it means breaking a formula down to its fundamental components and seeing how it's derived step-by-step from the most basic concepts (like definitions and axioms).

That might still sound a bit abstract, so let's take a simple example: the formula for the area of a circle, S = πr².

Most people's learning method (rote memorization): The teacher says, "Remember, the area of a circle is S equals pi times r squared." You nod, say "oh," and start solving problems, plugging in the radius, calculating the area, and you're done. As long as you remember the formula for the next exam, you're fine. But what if you forget it one day, or the problem changes form? You might get stuck. Deep down, you feel uncertain because you don't know why it looks that way.

Learning by 'getting to the bottom of it': When you see this formula, a few questions pop into your mind:

1. Why is it r squared? Why not r cubed? Or just r?
2. Where exactly does this π come from? What's its inherent connection to the area?

Alright, with these questions in mind, let's 'return to the origin':

Step 1: Return to the most basic concepts. What is 'area'? It's the size of the space inside a shape. What's the area formula we're most familiar with? The area of a rectangle = length × width. This is one of our 'origins,' a piece of knowledge we fully understand and don't need to prove.

Step 2: Find a way to connect the 'new problem' with the 'origin.' A circle is curved, a rectangle is straight. How do we connect them? This requires a bit of imagination, which is where math gets fun.

Imagine you cut a pizza (a circle) into many, many tiny slices (say, hundreds of them). Each slice looks like an extremely thin triangle.

Now, arrange these slices like teeth on a comb, half pointing up and half pointing down.

When you cut them finely enough, what shape does the arrangement increasingly resemble? — A rectangle!

Step 3: Analyze the parameters of this 'new shape.' For this approximate rectangle:

• What is its height? It's the side length of the original pizza slice, which is the circle's radius r.
• What is its length? It's the sum of the arc lengths of all the small sectors pointing up (or down). You've divided the entire circumference of the circle, giving half to the top and half to the bottom. So, the length of this rectangle is half of the circle's circumference.

Step 4: Calculate using the known 'origin.' We know:

• The formula for the circumference of a circle is C = 2πr (this could be another starting point for 'getting to the bottom of it,' but let's assume it's known for now).
• Therefore, the length of this rectangle is (2πr) / 2 = πr.
• The height of this rectangle is r.

So, the area of the rectangle = length × width = (πr) × (r) = πr².

Conclusion: You see, starting from the most basic knowledge point, 'the area of a rectangle,' and through the simple actions of 'cutting' and 'rearranging,' we've step-by-step 'constructed' the formula for the area of a circle.

After going through this process, when you look at the formula S = πr², it's no longer just a bunch of cold symbols in your eyes.

• You see the infinitely divided circle.
• You see the rectangle, formed by countless small sectors, with a length of πr and a height of r.
• You understand how π and r² 'collaborate' to express the internal space of a circle.

At this point, you can say you truly understand the formula. You can even re-derive it yourself by mentally reviewing this process if you ever forget the formula. Doesn't that feel much more solid?

To summarize, how to understand formulas using the 'getting to the bottom of it' method:

1. Don't fear it, question it: When you encounter a new formula, don't rush to memorize it. Ask yourself a few 'whys.'
2. Return to the origin: Think about the most basic definitions involved in this formula. What's the most familiar, related knowledge you have? (e.g., from the area of a rectangle to the area of a circle)
3. Act + Imagine: Draw it out, or build a mental model (the pizza cutting is a good example), and find a way to transform the 'unknown' into the 'known.'
4. Derive it yourself: Follow textbooks or videos and personally walk through the derivation process. The key isn't to memorize the derivation, but to understand the logic behind each step — 'Why is this being done?'
5. Build connections: Think about what other knowledge points this formula can be linked to. For example, once you understand the area of a circle, consider the volume of a sphere. Do they share a similar 'division' concept?

This process might feel a bit slow and 'laborious' at first, but trust me, once you get used to this way of thinking, the doors to the world of mathematics will truly open for you. It will no longer be just a tool for exams, but a playground full of logical beauty and creative joy.