The Perils and Rewards of Calculus: A Journey Through Understanding
It's been a long time, but when I was in high school, I found myself grappling with a calculus and solid geometry course. As a capable student in math, I managed to navigate through, albeit with a fair share of doubt about my true understanding.
My calculus experience at university was quite different. The first semester was a 'proofs' class, where theorems and subsequent methods of solving calculus problems were rigorously explored. For most of that semester, it felt like a strenuous climb—figuring out what delta-epsilon proofs entailed was a challenge. The subsequent semesters of the calculus sequence were a lot easier. Why? Simply because that first semester's struggle forced me to delve deeply into the subject, laying a robust foundation that made it smoother in the long run. Not only did four of my classmates from that initial course top the next semesters, but we credited the rigorous start for our success in subsequent courses.
The truth is, learning mathematics, particularly calculus, is always a step-wise process. While some students find one step more challenging than others, such as moving from algebra to calculus, some students hit a brick wall and are unable to proceed. Conversely, there are those for whom every new step is a triumph, leading us to call them mathematicians. I fall closer to the average student, not quite a mathematician, but one who appreciates the journey.
The Diversity of Calculus Classes
The nature and difficulty of calculus classes can vary greatly, depending on where, what, and how you are learning the subject. Some classes are taught with a rigorous emphasis on proofs, while others focus on intuitive understanding, skipping the proofs and focusing on simple computation.
In high school, I encountered a college-level calculus student who warned me that calculus would be incredibly challenging. It was a bit daunting, to say the least. My own calculus experience was different, as I was part of the MPC Math/Physics/Chemistry elite program. The class was designed for the top math students from various high schools. The professor warned us that 1/3 of us would fail, and he proceeded to teach a version of calculus that ensured that prediction would come true. Those who failed often switched to a math major or minor, excelling in their studies.
Teaching and Learning Styles
The level of difficulty in teaching calculus depends on the students and the methods used. Some classes might consist of just 30 students, while others could carry 650. The course level varies based on the intended field of study (math/physics, business, biology, social studies) and the students' preparation levels. In a scenario at a certain unnamed university, known for its relaxed standards, the course was relaxed, catering to the needs of the students present.
To excel in calculus, it's important to be well-prepared and not to fall behind. If something is not easily grasped, revisit the basic algebra and geometry behind it. Nearly all students find the calculus itself manageable, but the challenge lies in the necessary background information they may not fully understand. Over the years, millions of students have studied calculus, and those who passed likely reflect that it was relatively easy in retrospect. However, those who struggled may not fully grasp why they found it challenging.
Calculus, like any other complex subject, is both a journey of hardship and triumph. It is a subject that requires dedication, preparation, and a willingness to explore the basics before delving into the more intricate aspects. Whether you find calculus easy or hard, the journey is rewarding, and every understanding gained is a step towards mastering it.
The Conclusion
Calculus, while challenging, is indeed a journey filled with both challenges and rewards. By understanding the steps required, finding the right support, and embracing the foundational concepts, even those who find calculus initially daunting can grow to appreciate and master it. This journey is a testament to the power of perseverance and a solid educational foundation.
For those ready to embark on this journey, we offer resources, guidance, and a supportive community to help you navigate through the complexities of calculus. The road may be long, but the destination is worth the effort.