Do you really want to get better at mathematics?
Remember when you first learned how to program? I do. I spent two years experimenting with Java programs on my own in high school. Those two years collectively contain the worst and most embarrassing code I have ever written. My programs absolutely reeked of programming no-nos. Hundred-line functions and even thousand-line classes, magic numbers, unreachable blocks of code, ridiculous code comments, a complete disregard for sensible object orientation, negligence of nearly all logic, and type-coercion that would make your skin crawl. I committed every naive mistake in the book, and for all my obvious shortcomings I considered myself a hot-shot programmer! At least st I was learning a lot, and I was a hot-shot programmer in a crowd of high-school students interested in game programming.
Even after my first exposure and my commitment to get a programming degree in college, it was another year before I knew what a stack frame or a register was, two more before I was anywhere near competent with a terminal, three more before I fully appreciated functional programming, and to this day I still have an irrational fear of networking and systems programming (the first time I manually edited the call stack I couldn't stop shivering with apprehension and disgust at what I was doing).
In a class on C++ programming I was programming a Checkers game, and my task at the moment was to generate a list of all possible jump-moves that could be made on a given board. This naturally involved a depth-first search and a couple of recursive function calls, and once I had something I was pleased with, I compiled it and ran it on my first non-trivial example. Low and behold (even having followed test-driven development!), I was hit hard in the face by a segmentation fault. It took hundreds of test cases and more than twenty hours of confusion before I found the error: I was passing a reference when I should have been passing a pointer. In particular (and this is the aggravating part, as most programmers know), the fix required the change of about 4 characters. Twenty hours of work for four characters! Once I begrudgingly verified it worked (of course it worked, it was so obvious in hindsight), I promptly took the rest of the day off to play Starcraft.
Of course, as every code-savvy reader will agree, all of this drama is part of the process of becoming and strong programmer. One must study the topics incrementally, make plentiful mistakes and learn from them, and spend uncountably many hours in a state of stuporous befuddlement before one can be considered an experienced coder. This gives rise to all sorts of programmer culture, unix jokes, and reverence for the masters of C that make the programming community so lovely to be a part of. It’s like a secret club where you know all the handshakes. And should you forget one, a crafty use of awk and sed will suffice.
Now imagine someone comes along and says,
“I’m really interested in learning to code, but I don’t plan to write any programs and I absolutely abhor tracing program execution. I just want to use applications that others have written, like Chrome and iTunes.”
You would laugh at them! And the first thing that would pass through your mind is either, “This person would give up programming after the first twenty minutes,” or “I would be doing the world a favor by preventing this person from ever writing a program. This person belongs in some other profession.” This lies in stark opposition to the common chorus that everyone should learn programming. After all, it’s a constructive way to think about problem solving and a highly employable skill. In today’s increasingly technological world, it literally pays to know your your computer better than a web browser. (Ironically, I’m writing this on my Chromebook, but in my defense it has a terminal with ssh. Perhaps more ironically, all of my real work is done with paper and pencil.)
Unfortunately this sentiment is mirrored among most programmers who claim to be interested in mathematics. Mathematics is fascinating and useful and doing it makes you smarter and better at problem solving. But a lot of programmers think they want to do mathematics, and they either don’t know what “doing mathematics” means, or they don’t really mean they want to do mathematics. The appropriate translation of the above quote for mathematics is:
“Mathematics is useful and I want to be better at it, but I won’t write any original proofs and I absolutely abhor reading other people’s proofs. I just want to use the theorems others have proved, like Fermat’s Last Theorem and the undecidability of the Halting Problem.”
Of course no non-mathematician is really going to understand the current proof of Fermat’s Last Theorem, just as no fledgling programmer is going to attempt to write a (quality) web browser. The point is that the sentiment is in the wrong place. Mathematics is cousin to programming in terms of the learning curve, obscure culture, and the amount of time one spends confused. And mathematics is as much about writing proofs as software development is about writing programs (it’s noteverything, but without it you can’t do anything). Honestly, it sounds ridiculously obvious to say it directly like this, but the fact remains that people feel like they can understand the content of mathematics without being able to write or read proofs.
I want to devote the rest of this post to exploring some of the reasons why this misconception exists. My main argument is that the reasons have to do more with the culture of mathematics than the actual difficulty of the subject. Unfortunately as of the time of this writing I don’t have a proposed “solution.” And all I can claim is a problem is that programmers can have mistaken views of what mathematics involves. I don’t propose a way to make mathematics easier for programmers, although I do try to make the content on my blog as clear as possible (within reason). I honestly do believe that the struggle and confusion builds mathematical character, just as the arduous bug-hunt builds programming character. If you want to be good at mathematics, there is no other way.
All I want to do with this article is to detail why mathematics can be so hard for beginners, to explain a few of the secret handshakes, and hopefully to bring an outsider a step closer to becoming an insider. So read on, and welcome to the community.
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