I would have to say the Math 134 series with John Palmieri was my favorite class of freshman year.
I always like math; at my small private high school without AP classes, because I could go faster than my classmates I studied math independently for the last year and a half. I learned a lot, but I had little to show for it in terms of course grades or test scores. So I was happy to learn there was an honors first-year calculus course that did not have any prerequisites. It was the equivalent of five quarters of math in three quarters, so it was an efficient head-start on my math major.
The series covers the material from Math 124, 125, 126, 307, and 308, but faster and with more theoretical depth. We began the class by discussing proofs, particularly mathematical induction. At first, I thought that mathematical proofs should be written as much as possible in mathematical symbols, like the symbolic logic proofs I had done in eighth grade. But actually, good proofs should include extensive verbal explanation, stating equations for key points, but avoiding unexplained series of equations. Even mathematicians need equations explained in words.
But this takes nothing away from mathematical rigor. In fact, rigor is one of the goals of the course. Before citing any theorem, you must make sure all the hypotheses are satisfied. This is part of the reason we covered more theoretical concepts like the definition of the real numbers, domains and codomains of functions, epsilon-delta limit proofs, proofs of limit properties, proofs of properties of continuous functions, proof of properties of differentiable functions, and proofs of everything else. Just reading over that list of delicious math made me lick my lips.
There is also mathematical humor. Around Thanksgiving, we applied Newton's Law of Cooling to a Turkey (the problem was taken from James Stewart’s textbook, which is used in 124). A turkey was taken out of the oven at 185 degrees F and set in a room at 75 degrees. After 30 minutes, the turkey is at 150 degrees. What is the temperature after 45 minutes? When is the turkey at 100 degrees?
Newton’s law of cooling is an inaccurate model for a turkey, the professor said, because it is not at a uniform temperature, and when it is first taken out of the oven, it is actually hotter near the edge than in the middle. The problem is also inaccurate because the turkey wouldn’t just sit there with nobody eating it. But in fact you should never cook a turkey at 185 degrees; it would turn to sawdust! So maybe it was accurate that no one ate the turkey!
The professor did the problem anyway. “And the moral of the story is, don't learn to cook from Stewart.”
In Spring, for Math 136, we wrote a short paper in teams of two or three about some application of linear algebra. Ours about cryptography. I will post it soon.