How can First Principles help students overcome the pitfalls of 'rote problem-solving'?

Haha, that's an excellent question, and it's a dilemma many people face. Let me explain it to you with a real-life example, and you'll grasp it immediately.

"Drill practice" (or "rote learning") is like memorizing cookbooks.

You buy a thick cookbook filled with recipes like "Fish-Fragrant Pork Shreds," "Kung Pao Chicken," "Mapo Tofu," etc. You memorize every ingredient and step for each dish by heart. If someone asks you, "How do you make Fish-Fragrant Pork Shreds?", you can answer flawlessly. As long as the dish is in the cookbook, you can make it.

But here's the problem:

If you're asked to make a dish not in the cookbook, you're instantly stumped. For example, if you're told to make "Coke Chicken Wings," and it's not in your cookbook, you wouldn't know what to do.
If you're missing an ingredient in the kitchen, you wouldn't know how to adapt. For instance, the recipe calls for "Pixian Doubanjiang," but you only have regular chili paste. You'd hesitate because you don't understand that the core function of Pixian Doubanjiang here is to provide saltiness and spiciness, and it can actually be substituted.
It's extremely inefficient. There are countless recipes in the world; you'll never finish memorizing them all.

This is the dilemma of "drill practice": you're just mechanically memorizing solutions instead of truly understanding. When you encounter a new type of problem, or an old problem phrased differently, you're completely lost.

"First principles thinking," on the other hand, is like learning the underlying logic of cooking.

Instead of rote memorizing specific recipes, you seek to understand:

The basic principles of taste: What are salty, sweet, sour, spicy, and umami? How do they balance and complement each other? (e.g., sugar enhances flavor, vinegar cuts grease)
The physical and chemical changes of ingredients: Why marinate meat first? (To absorb flavor and make it tender) Why stir-fry quickly over high heat? (To lock in moisture and keep it crisp) What is the Maillard reaction? What is caramelization?
The significance of knife skills and heat control: How do different cuts of ingredients affect texture? What cooking scenarios are best suited for low, medium, and high heat, respectively?

Once you've mastered these most basic, core "principles," what happens?

You can create new dishes. Given chicken wings and a bottle of Coke, you'd think: chicken is meat, it needs flavor; Coke is sweet, and heating it will make it viscous. So, I can combine them to create "Coke Chicken Wings." You don't need a recipe because you can "derive" one yourself.
You can adapt flexibly. If you don't have cooking wine, you know you can use white wine or beer instead, because their core function is to remove fishiness and add aroma. If you don't have soy sauce A, you can use soy sauce B, because you know the core role of soy sauce here is to provide saltiness and savory flavor.
Learning efficiency is extremely high. You don't need to memorize 1000 recipes; you only need to master dozens of core principles to handle thousands of ingredient combinations.

Bringing this back to learning: Learning with "first principles" isn't about solving 100 problems on "parabolas"; it's about truly understanding: What is the definition of a parabola? Where does its equation come from? What do fundamental elements like the focus and directrix actually mean?

Once you've thoroughly grasped these foundational axioms, definitions, and theorems, any problem about parabolas will simply appear as a combination of basic elements to you. You'll no longer need to memorize "this problem type uses method A" or "that problem type uses approach B"; instead, you can calmly take out your most basic tools and "assemble" a solution path on the spot.

In essence, first principles thinking transforms you from a "skilled problem-solver" into an "engineer who understands the principles." It empowers you to solve problems you've never encountered before, which is the true purpose of learning and the only way out of the bottomless pit of "drill practice."