I have loved math for as long as I can remember. In elementary school I was perfectly content to do worksheets full of practice problems; in middle school I participated in my first math competition and was hooked on combinatorics and analytic geometry; in high school I spent countless hours poring over my math textbooks and exploring the art of mathematical proof. Before I registered for autumn quarter classes, an advisor I talked with suggested I look into the MATH 134 sequence; I researched the course, decided it sounded like my cup of tea, talked to the math advisor about my background and interest in the course, and was fortunate to get in. My experience in the course has been extraordinarily positive, and I have absolutely no regrets about my choice to take it.
The class hit the ground running on Day 1. The professor, Dr. Don Marshall, was very clear from the beginning about the philosophy of the course: we were expected to do the reading and associated problems before class and bring in questions about anything that confused us, so we could focus on clearing up misunderstandings before moving briskly on to the next section of material. He explained that he wanted everyone to succeed and if the format of the class sessions wasn’t working he was happy to make changes.
Soft-spoken and even-tempered, with chalk dust on his elbow from habitually holding his arm with chalk still in hand, Professor Marshall listens to his students with an intensely intelligent expression and cracks an amiable grin when speaking lightly about the math we’re discussing. He treats the students with respect and holds us to high standards of academic performance while doing everything he can to help us rise to those standards.
Maintaining that level of academic performance is not easy. Let me emphasize: this class is a lot of work. With substantial textbook readings and practice problems mostly every evening and also over the weekends, weekly homework and quizzes, and an emphasis on deep understanding and thorough proof, the class requires a serious sustained effort just to keep pace. When they named it “Accelerated Honors Calculus,” they really weren’t kidding.
My classmates are all working hard, just as I am, and it’s not for lack of mathematical aptitude. On Day 1, Professor Marshall quoted a colleague as he described the learning environment: “Look at the two people sitting next to you. One of them is smarter than you are, and the other is better prepared.” Taken literally, this implies a spontaneously emerging seating arrangement with increasing intelligence and decreasing preparedness, but the point was well taken. It’s common for my peers to chatter earnestly before class about conditions of Riemann integrability or the hyperbolic trigonometric functions. And yet amidst the heavy workload of the class, there’s always a sense of good humor in the room: my peers are as fully capable of conversing about uniform continuity as they are of engaging in an animated mock debate about whether one can apply Cartesian existentialism as a narcissistic approach to many-worlds theory. (Long story.)
Being fond of precise reasoning and elegant solutions, I enjoy the course’s focus on proof and on building up a rich library of theorems with no mathematical hand-waving. At the same time we aim to develop an intuitive geometric understanding for what the equations are saying, building up our ability to translate big-picture concepts into the language of math. The homework assignments require conceptual understanding in order to arrive at a solution and the skill of rigorous proof to write up a detailed solution. In the words of the TA, Austin Stromme: “Just make sure you’re justifying every step.” This is of course easier said than done, and I learned the hard way early on that if a function must be continuous for a theorem to apply, one has to explicitly mention that fact before invoking the theorem, even if it seems too obvious to be worth stating.
Here’s a representative homework assignment, showing the kind of proofs that we practice.
The intuitive ideas of calculus are beautiful, and the rigor makes makes the ideas meaningful. This class pushes me to improve both my mathematical intuition and my ability to write the results of this intuition rigorously. I’m finding the endeavor extraordinarily worthwhile and I’m looking forward to continuing in the series.