This semester I have the privilege/responsibility of teaching a class whose students, all very intelligent and highly motivated, are trying to get a jump start on higher level mathematics. The course is advertised as Multivariable Calculus "honors" course. I hesitate to even call it a course in multivariable calculus, as we won't cover any of that material until the second half of the course. The first half of the course will be laying a very theoretical groundwork for what students in other classes are already learning. Our class will have the advantage of seeing these concepts the "right" way--understanding the derivative in terms of linear functions, understanding functions in a rigorous way, and understanding vectors and vector spaces abstractly.
This week I am focusing on function theory and set theory. That means proofs--and lots of them. One of my students in office hours was complaining as much. (My office hours, by the way, had to be held in the common room in the math building, because apparently my office must now be under construction until an unspecified time. There was no notice of this occurrence before it started, and those of us who occupy said office still have no idea when it will end. Also, today I went in my office to see if it was finished, and I noticed that someone had messed with my Rubick's cube! All of the stickers had been taken off! Who messed with my stuff, anyway? But I digress...)
So, proofs... Yes, entering the world of mathematical rigor is not easy for most students. I have to be honest, it was pretty easy for me when I first started seeing proofs in college. Part of that was that I had a proof-based course in high school through the EPGY. Another part of it is that I've always been a naturally skeptical person, and very particular about the meaning of statements. For instance, once in fourth grade I actually battled my entire class, including my teacher, over the statement, "A square is a rectangle, but a rectangle is not a square." Obviously, the correct statement would be, "Not all rectangles are squares," but the subtlety was lost on my peers, and, yes, even my teacher, as far as I can tell to this day.
Proof writing is a beautiful and delicate art. The truly great mathematical writers are the ones who get beyond just the brute calculations and really explain what's going on. Often people ask me how one actually does research in mathematics. The answer is the same as in any other field; you come up with a new idea and you present it. This is done on a scale from acceptable to great. Acceptable mathematics presents true statements and sufficient details to prove those statements. Great mathematics presents not only true statements, but powerful ones; and it doesn't just prove the results, it also expounds the key ideas that make the results true. In other words, great mathematics is great writing, just as in any other field.
What is unique to mathematical writing is the notion that even beautiful ideas must eventually pass the test of correctness. I say "eventually" because in all truth, many beautiful ideas are worth presenting even if they're wrong. Nevertheless, even beauty is not nearly as subjective in this field as it might be in others. "Beauty" usually corresponds to genuine insight, just like a "beautiful runner" is also probably a fast one.
That's why I think all students could really benefit from taking a course that demanded mathematical rigor. When I suggest that students not be required to take calculus, it's actually because I think they should be asked to do more, not less. I'd much prefer that students take a course in some of the fundamentals of mathematics, in which proof writing was an essential part of the curriculum, than learn a bunch of formulas for differentiating functions. Learning to prove things is exactly what it sounds like--if everyone knew how to do this, we'd have a lot less nonsense in the world.
Here are the common hurdles students usually have to overcome when they're first being introduced to mathematical rigor. The first is understanding that concepts in mathematics have extremely literal meanings. Although you generally have to learn metaphorically (such as using graphs to understanding functions), eventually you have to apply the literal meanings in order to prove theorems. This can be really hard. Intuitive arguments sound plausible at first, but a professional skeptic will be able to quickly tear them down. In a sense, all mathematicians are professional skeptics.
The literalness of mathematical concepts means that a lot of proofs can be done by rote, once a general method has been ingrained in students. For instance, to show a function f:A-->B is surjective, every proof can go like this: "Suppose b is in B. Then something about the function implies that........and so, there exists an a in A such that f(a) = b. QED" For an experienced student of mathematics, this is mind-numbingly obvious. For a beginner, it is a mysterious incantation. The really good students will quickly realize how to construct a proper framework for proofs based on the statement being proved. The others will simply have to punish themselves with repetition until the patterns sink in.
The second hurdle represents is an equal but opposite difficulty: mathematics takes insight. Most students will tell me at one point or another during this course, "I don't know where to start." Of course you don't. That's why it's called a "problem." If all problems in mathematics had obvious starting points, there really would be no such thing as research in mathematics--and this is probably why the average person does not realize that mathematics still remains to be discovered.
I find this hurdle difficult to deal with as an instructor. As a calculus teacher (particularly in the "121" class I taught) I never had any qualms about just spoon-feeding my students a strategy for solving the problems that were going to be on the exam (although they did not perceive it that way). For this course, I feel I have to be relatively austere, like a parent finally telling his children, "Well, you're on your own." Real math problems, like life, don't have easy answers. Sometimes no one knows where to start. That's not true of any of the problems my students will deal with this semester, but that's not the point. The point is that in order to really learn mathematical reasoning, you have to be willing to battle through the process of discovery. You have to earn that theorem, if you really want to know what it says.
The third hurdle, the one I intend on addressing when I teach tomorrow, is hurdle of sophistication, for lack of a better term. It's one thing to answer a question. It's quite another to really own it. David killed Goliath with one stone. That's what a real mathematician wants to do; there's no sense writing out a bunch of brutal computations when a clever trick will do.
For instance, on a quiz just turned in my students were asked to find a function f:N-->N, where N is the natural numbers, which is neither injective nor surjective. I think a total of three students in the class really "got" the problem. Don't get me wrong; most of them got full credit for the problem. But only a few students understood how much freedom they had, and therefore how easy it was, to bury this problem in the ground. Sure, it's natural for students at this point to construct such a function by modifying n^2 somehow to make it no longer injective, or to come up with some double counting strategy which nevertheless misses 1. But real mathematical skill is seen in the beautifully simple response, "f(n) = 1." That's the stone that killed Goliath. No explanation is even required. (Don't feel bad if you don't understand; but you could try looking here and here if you want to know what I'm talking about.)
That's what I think it means to have mathematical style. It means you feel free to defy convention, to let yourself be restricted only by logic and your own creative insight. It means you don't throw down a page of computations when three lines of cleverness will do. It means you don't just know mathematics, you own it. I don't think I can actually teach this. The best I can hope for is that some students will find me to be a helpful guide through their own mathematical journey.