Today is positive, I figured out the real meaning of studying. The talking and thinking are the following:

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The difference you describe is that between learning-by-recipe and learning-by-purpose.

Learning-by-Recipe

This is the traditional, and often flawed, way that many educational systems operate. It focuses on the "what" and the "how," but not the "why." You are given a problem and a predetermined method to solve it. For example, "Here is a quadratic equation; use the quadratic formula to find the roots." The link between the problem and the solution is rigid and often lacks context.

When learning this way, our minds create a direct binding as you mentioned: Problem A = Solution A. This is efficient for standardized tests but terrible for real-world application, where problems are rarely so neatly packaged. It can make you feel like you have to know all the formulas and solutions upfront, which is overwhelming and stifles creativity.

Learning-by-Purpose

This is the approach you're advocating for, and it's far more effective for deep understanding and innovation. It starts with an internal motivation to solve a problem. The process looks more like this:

    Identify a problem that you genuinely care about.

    Explore the problem yourself, trying to formalize it, break it down, and perhaps even propose a preliminary solution based on your existing knowledge.

    Search for better ways to solve it. This is where you actively seek out and learn the "methods" and "formulas" that others have developed.

The key difference here is that you are not a passive recipient of information; you are an active investigator. You are not just learning a solution; you are learning how that solution works, its strengths, and its weaknesses relative to the problem you care about.

Also, I persistently develop Rust programs. And it’s getting more and more promising to build scientific computation library in Rust, as I’m learning about mathematics and physics.