1. The Allure of the Advanced vs. The Strength of Foundations

Firstly, nearly every budding mathematician (pure or applied), physicist, natural scientist, economist—indeed, anyone with an academic spirit—eventually faces a common dilemma: Do I chase the thrill of understanding the latest, greatest advancements in my field, perhaps jumping in over my head? Or do I patiently take the time to truly solidify my foundational knowledge?

This is a particular challenge many gifted high school students encounter. On one hand, they’re often encouraged to master foundational subjects like linear algebra and real analysis. On the other, there’s that undeniable thirst for a bigger challenge, for something more exotic and cutting-edge. So, it’s not uncommon for them to be drawn towards reading about grand topics like the Poincaré Conjecture, the Riemann Hypothesis, or Fermat’s Last Theorem, sometimes at the expense of the basics.

I’d be less than honest if I didn’t admit to being in that exact position myself at one point. However, it’s incredibly important for any aspiring mathematician or academic to cultivate a “sense of measure”—to know when to pause the chase for the complex and circle back to reinforce the fundamentals.

For instance, one such exciting, albeit advanced, overview I encountered was Evan Chen’s ‘An Infinitely Large Napkin’ (which I also mention in my Book Recommendations). While it’s a fascinating and illuminating read, it’s crucial to balance such explorations with dedicated time spent working through core material from established textbooks. Simply put, it can be counterproductive for even the most mathematically gifted mind to flit from one advanced topic to another, accumulating a superficial understanding of many things but a deep, satisfactory knowledge of few.

A far more productive and sustainable approach involves methodically reading various foundational textbooks and diligently working through a significant portion of the exercises. Yes, it takes longer, and it requires discipline. But the reward is an unshakable, deeply rooted foundation upon which all future learning can be confidently built.

Okay, let’s refine that section on studying for proof-based courses to make it a tiny bit less formal and more concise, while keeping it very polite and respectful.

2. Finding Your Footing in Proof-Based Math 🧭

Jumping into proof-based math courses, like abstract algebra or real analysis, is a pretty big shift from what you might be used to. It’s less about just crunching numbers and much more about carefully building logical arguments. This definitely calls for a fresh way to study and think.

First off, getting a real grip on the theorems is key. This means going beyond just memorizing what they say; digging into how they’re proven is where you’ll really start to understand things deeply. When you study the proofs from class and your textbook, you start to get a “feel” for the common patterns and techniques used in that specific area of math. Once you truly know your theorems well—their conditions, their conclusions—you’ll have a much better toolkit for figuring out how to approach new problems.

Don’t underestimate those early problem sets either! They might seem simpler than what comes later, but they’re packed with the basic proof-writing tools you’ll need again and again (like direct proofs, how to use contradiction, or setting up an induction). Wrestling with these early on, even if it feels tough, is a vital part of learning. It’s how you get comfortable with methods that will become second nature. Along the same lines, your class notes, especially the proofs your professor chooses to emphasize, are super helpful for seeing what concepts and techniques are most important.

Learning how to write proofs isn’t something you usually have to do all alone. Talking things over with classmates or study partners can be a huge help. Trying to explain an idea to someone else is one of the best ways to make sure you understand it yourself, and you’ll pick up different ways of looking at problems too. If your university has a math tutoring center, definitely check it out for guidance when you’re feeling stuck. And don’t be shy about chatting with your professor during their office hours; they can offer great feedback and help you see how all the pieces fit together.

Now, a quick word on using AI and Large Language Models (LLMs) for help. They can seem like a handy tool when you’re stuck on a tough proof, but it’s good to be a bit careful. For complex math, AI-generated solutions can sometimes have hidden mistakes or logical gaps, even if they look right at first glance. Plus, a really big part of learning proofs is the “struggle” itself — that process of working through it, trying different things, is what helps you learn deeply. Relying too much on AI can mean missing out on that important experience.

My friendly advice? Try talking to actual people first. Or, you could always try the classic programmer’s trick: explain your proof, step by step, to an actual rubber duck 🦆! You’d be surprised how often just articulating the problem and your reasoning aloud helps your own mind find the solution. Ultimately, it’s your call how you use these tools, but focusing on building your own strong reasoning and problem-solving skills is what will really count in the long run.