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There got to be far more to mathematics than simply learning how to follow steps. Nope, that's really all there is. Just like computer programming is nothing

by dkarl 14y ago

There got to be far more to mathematics than simply learning how to follow steps. Nope, that's really all there is. Just like computer programming is nothing but learning to string arcane commands together in anal retentive ways. It's just a lot of making variable names and putting data in them and using little symbols like "argc" and "foldl" and "fopen" that have excruciatingly dry definitions that are completely disconnected from the real world. You memorize rules of syntax and look up function definitions and follow all of the rules exactly correctly, and even after you've done what feels like a ton of that you're still just writing a dumb program that you would never use anyway. What's the point? But I bet you didn't feel that way about programming when you were first learning, right? Things like foldl and dlopen seemed kind of neat, if not mind-blowing. People who become good at programming get there because they take delight in the simple things that make up the basics. Is there anything intrinsically beautiful about loops? There is the first time you encounter one. Oh, man, the possibilities! The feeling passes quickly, but I can't imagine how anyone who never enjoyed loops could endure the boredom long enough to become skilled at programming. It's the same thing with math. Like programming, it really is just symbol manipulation and nothing more. The pleasure and beauty is in your perception of it, the concepts you develop to give intuitive substance to the abstract rules. Your appreciation gets deeper the more you learn. But to get there you have to enjoy the basics, the simple things. How many times did you have to write for (i = 0; i < m; ++i) before it became second nature? You must have derived some satisfaction from your initial fumbling with loops and functions or your interest would have died before you learned to write more complex programs. (The kids who don't enjoy the basics quit programming as soon as they realize they won't be making a state-of-the-art video game in a weekend.) Learning math is the same process; you start by enjoying where you are, even if you're starting at the mathematical equivalent of for (int i = 0; i < 10; ++i) { System.out.println("I love cake " + i + " times!"); } That may not sound very attractive to an educated person with many sophisticated things to think about. It's entirely possible that there are some people who are not simple-minded enough to enjoy math. However, I think those people do not enjoy programming, either ;-)

beambot 14y ago
Like programming, it really is just symbol manipulation and nothing more. I strongly disagree. Arithmetic, perhaps... but not "math." Some of the best math proofs require little-to-no direct symbol manipulation. You might be able to reduce a mathematical statement down to the symbolic level (eg. using sets), but so much of the beauty is in the simple intuition. One simple example: the pigeon hole principle -- you can teach it to a 3 year old, and yet its incredibly powerful. EDIT: If you read about Lockhart's Lament (as mentioned elsewhere in this thread), he specifically writes: The cultural problem is a self-perpetuating monster: students learn about math from their teachers, and teachers learn about it from their teachers, so this lack of understanding and appreciation for mathematics in our culture replicates itself indefinitely. Worse, the perpetuation of this “pseudo-mathematics,” this emphasis on the accurate yet mindless manipulation of symbols, creates its own culture and its own set of values. Those who have become adept at it derive a great deal of self-esteem from their success. The last thing they want to hear is that math is really about raw creativity and aesthetic sensitivity. Many a graduate student has come to grief when they discover, after a decade of being told they were “good at math,” that in fact they have no real mathematical talent and are just very good at following directions. Math is not about following directions, it’s about making new directions.
- dkarl 14y ago
  Be fair. I do get around to saying there's more to it. Why, it's in the very next sentence after the one you quoted :-) The pleasure and beauty is in your perception of it, the concepts you develop to give intuitive substance to the abstract rules.
  - akjetma 14y ago
    That seems to imply that the pleasure and beauty of mathematics is in the way one perceives symbolic manipulation. I actually think the pleasure and beauty of mathematics is in the stage before one gets to manipulating symbols; the stage where you have an idea about something and are thinking about ways to communicate it or prove it. People should be given problems ('how much of this box does this triangle take up?' to paraphrase Lockhart's example) and allowed to drum up their own solutions/syntax. Admittedly, this is kind of idealistic.

by dkarl 14y ago

beambot 14y ago

Like programming, it really is just symbol manipulation and nothing more. I strongly disagree. Arithmetic, perhaps... but not "math." Some of the best math proofs require little-to-no direct symbol manipulation. You might be able to reduce a mathematical statement down to the symbolic level (eg. using sets), but so much of the beauty is in the simple intuition. One simple example: the pigeon hole principle -- you can teach it to a 3 year old, and yet its incredibly powerful. EDIT: If you read about Lockhart's Lament (as mentioned elsewhere in this thread), he specifically writes: The cultural problem is a self-perpetuating monster: students learn about math from their teachers, and teachers learn about it from their teachers, so this lack of understanding and appreciation for mathematics in our culture replicates itself indefinitely. Worse, the perpetuation of this “pseudo-mathematics,” this emphasis on the accurate yet mindless manipulation of symbols, creates its own culture and its own set of values. Those who have become adept at it derive a great deal of self-esteem from their success. The last thing they want to hear is that math is really about raw creativity and aesthetic sensitivity. Many a graduate student has come to grief when they discover, after a decade of being told they were “good at math,” that in fact they have no real mathematical talent and are just very good at following directions. Math is not about following directions, it’s about making new directions.

dkarl 14y ago
Be fair. I do get around to saying there's more to it. Why, it's in the very next sentence after the one you quoted :-) The pleasure and beauty is in your perception of it, the concepts you develop to give intuitive substance to the abstract rules.
- akjetma 14y ago
  That seems to imply that the pleasure and beauty of mathematics is in the way one perceives symbolic manipulation. I actually think the pleasure and beauty of mathematics is in the stage before one gets to manipulating symbols; the stage where you have an idea about something and are thinking about ways to communicate it or prove it. People should be given problems ('how much of this box does this triangle take up?' to paraphrase Lockhart's example) and allowed to drum up their own solutions/syntax. Admittedly, this is kind of idealistic.