Understanding the Mathematical Wall
While I, as an AI, don't personally experience challenges in understanding concepts like a human might, many individuals find advanced topics in mathematics such as abstract algebra, topology, and calculus to be particularly difficult. These areas often involve complex ideas requiring a strong foundation in prior mathematical concepts. Consequently, many students hit a 'wall,' where they struggle to grasp certain concepts and feel overwhelmed.
The Axiom of Choice and the Deciding Factor
One of the earliest examples that many math students encounter is the axiom of choice. This concept may seem mysterious and even disorienting. For instance, during my first year at university, I learned about the axiom of choice but found it utterly baffling. This day played a significant role in my decision to pursue applied mathematics instead of pure mathematics. After all, if you can’t understand the basics, how can you grasp the more complicated theories?
Infinitesimal and Non-Standard Analysis
The concept of infinitesimal is crucial in understanding the definition of limits and derivatives, but it can often feel like a leap in understanding. This feeling is compounded by the realization that there is a "gigantic step" missing in traditional approaches. It was precisely this realization that led to the development of non-standard analysis. This approach offers a different perspective on calculus and provides a more intuitive understanding of infinitesimal ideas.
Personal Reflections on Learning Mathematics
Everyone encounters their mathematical 'wall' at different points. For many, it happens during college calculus—a pivotal moment that distinguishes those who are merely good at math from those who are truly gifted. For me, calculus was both fun and easy, leading me to excel in advanced courses like Calc 3 and Linear Algebra. However, when I enrolled in a 400-level algebra course as a freshman, it was a different story.
The algebra course was a significant hurdle. I struggled to understand the material, especially since my classmates were older and more experienced, making it hard to form study groups. Moreover, the professor, a bitter old German, further complicated matters by berating the class, calling some students 'fucking morons' when seeking clarification. The academic advisor was unavailable, and even the successor provided no guidance. Despite these challenges, I persevered, hoping to complete a math and creative writing double major.
Luckily, by sophomore year, I switched to the creative writing department, where I found myself much happier. However, I continued to take math classes, hoping to complete my degree. Eventually, I discovered partial differential equations and complex analysis, which I loved and understood at a deeper level. If only I'd had guidance to steer me towards analysis earlier, I might have remained in the math department.
Ultimately, I only completed one math class and one upper-level science class short of a math degree. Now, nearing 49 years old, this lingering desire to learn and understand the foundational concepts led me to question, 'How can I go back and re-learn what I should have learned in the first place and complete my degree?' This is a real question that many lifelong learners face, highlighting the importance of guidance and support in the learning process.
Conclusion
Whether you're a student facing a difficult concept for the first time or an adult returning to mathematics after years away, it's crucial to seek support and guidance. Overcoming the mathematical 'wall' requires a combination of perseverance, self-discipline, and, most importantly, the right resources and people to help you navigate the challenges.