If Math is a language, then the first 10 years of instruction are essentially spelling tests. There is very little flexibility nor room for creative thought in those years of instruction. Most kids never get to a proof or other areas without a defined path from start to finish. That is like only studying English grammar and never taking a literature test. Sure it can be interesting, but it is hardly creative.
I see mathematics, especially the more abstract areas, as much more similar to the arts than to the sciences.
I would seriously have loved doing proofs in high school. I doubt it occurred to any of our teachers, or if they had any understanding of this themselves. Now, my math skills are highly lacking due to dropping the subject too early and I would pay very good money for a decent online calculus class like the many great programming classes we're seeing pop up.
I had the good fortune of being taught via a theorem-proof method from my Junior year of High School onwards. It really did make a huge difference. Though I am a Computer Engineering major, I've taken some upper level math classes for fun and do significantly better than most students because I was taught proofs so early.
Proofs are boring. Who needs proofs?
You either know what you're proving is already proved, then what's the point?
Or you know it probably is false, then there's even less reason to.
People want math to be able to make things with it, all the math which doesn't help making things should be confined to 0,5% of population who really want this kind of punishment.
And responding on the title: the answers starts with "bo" and ends with "ring".
Let me paraphrase you: Knowing how to multiply two 3 digit numbers is boring. The method has been known for thousands of years, and you've got calculators that can do it for you. So why should one learn how to do that?
The answer is that it gives you intuition about how these things behave, even if you always use a calculator to come up with the correct result.
And mental back-of-the-envelope calculation is unbelievably useful in every day life; if you can't do it, I can guarantee that you're either making worse decisions for it, or wasting a lot of time optimizing your every purchase by typing it on a calculator.
Similarly, you need to practice proving and reading proofs so you actually have the right intuition about what constitutes "proof". Unfortunately, all of western culture, and american culture in particular, is so far divorced from the concept of sound logic, that it is very likely that most people don't even realize they are missing something.
It's not any of the specific proofs that matter. It's the exercise. You go to the gym to exercise your muscle; You do proofs to exercise your logic.
Because proofs are interesting. They're the "under the hood" of larger chunks of math knowledge. If you understand how to follow a proof, you'll have a much better understanding of the strengths and limitations of various mathematical statements. If you can work a proof youself (which is a lot harder actually), then you can participate in the global conversation among mathematicians and maybe even contribute.
For most people, dealing with mathematical proofs is a kind of parroting.
They learn a path and can follow it all right, but they never deviate from the said path because there's no point.
Which might be insightful but is inherently boring, because there's no way you'll make anything that didn't exist before. You only make some mind ways in your brain which will perhaps help you one day dealing with some real problem with real output.
Might as well flash a firmware over your brain.
Given the current state of math, very few people would ever be able to contribute anything.
Others can skip to the plan B immediately.
For most people, dealing with any math is parroting, which is the problem. Creating proofs of simple things is well within the grasp of bright high school students, and is a wonderful experience -- I feel sorry for anyone who misses out on it. Really, very little traditional mathematics is required to enjoy proving that there is no solution to the bridges of Konigsberg problem, no way to cover a chess board with dominoes leaving only opposite corners exposed, or no largest prime.
Most people don't learn proofs. If you're going to learn about proofs beyond mere knowledge that they exist, then you have to be forced to do proofs from scratch.
Doing a proof from scratch is akin to being given a compass and told that somewhere past the wilderness is the promised land where others have blazed trails to but otherwise given no other help.
You know you can create the proof (since its a book problem) but how to is entirely up to you.
guard-of-terra said You either know what you're proving is already proved, then what's the point? Or you know it probably is false, then there's even less reason to. Well I guess that's true of every kind of math a 12-year-old does. Why bother doing any of it? Well, mainly because it's a useful skill to have when you want to do something interesting.
Learning "useful skills" preemptively while doing no actual work is counterproductive. It would probably work if you could motivate yourself all along, but in the current world with all the distractions, you probably would not.
It's like learning to swim in an empty pool while being reassured that it would be filled once you're able to swim.
Programming is instantly addictive because you create from the day two. Math isn't.
Programming is instantly addictive because you create from the day two. Math isn't.
It isn't, because students aren't given problems to be solved. They're given the solutions first.
The profound feeling of figuring out a solution to a problem (a real one, unlike the ones given at school) is hard to explain to someone who did not experience it. That's why, having choice to do Masters degree in Math or CS, I choose Math.
Students are sad losers unless they practice programming before and even instead of college.
Programming is addictive, learned long before you decide to specialize in it, and you make up problems for yourself creating unique software products from day one.
With math, you only reiterate the same thing over and over.
That's why, having choice, I choose to work as a programmed early and do as little as possible without being kicked off at my higher education facility.
Oh no, wait - I did as little as possible even before I began working. I just amn't good at that learning thing perhaps.
With math, you only reiterate the same thing over and over.
If you do that, yes, it's dull and pointless. The thing is, you don't need to do it. It's like complaining that programming sucks because writing hello world over and over again is boring.
At this point I have to say we are just different people, I get those feelings from programming and not math for you it is vice verse. Even when the Math is programming related it still doesn't hit those cues. For me, my friend, happiness is a warm compiler.
Actually, I just had this conversation with a student of mine; I teach CS, and have been trying to throw at least a few proofs each term into most of my upper-level classes. They're generally simple proofs and the initial reaction is that the thing to be proven is "obvious" and thus doesn't need to be proven.
...but then, as they try to write out the steps of the "obvious" proof, or explain them to me verbally, we discover that they don't have nearly as good a handle on all the definitions and the details as they thought they did. Having to wrangle a proof, even a simple one, does an unparalleled job at developing a deeper understanding of the theory.
(For example, in my database theory class: Prove that any two-column table must be in BCNF. On the way through to the "trivial" two- or three-line proof, one has to understand what it means to be a key of a relation, what a functional dependency really is, and the notion of transitive dependency. Even if you never decompose a table into BCNF, the other concepts help you build some important mental models.)
Once again, you are referring to simple proofs. A spelling test is also boring.
What I am referring to is the level where math is a set of tools. Rather than doing algebra, I frequently finding myself create new algebras. Every time I create a new algebra (in order to model data, for example), I informally prove that it meets the required axioms to be the mathematical ring or group I mean it to be.
It's a whole different type of math. It's like reading shakespeare (where there are a bunch of difficult words) vs writing shakespeare (where a master of the English language invents new words to suit his purpose, while still conveying the message clearly).
Abstract Algebra is my favorite math topic. I was addicted after one semester. Proving things about groups and rings, and making up your own groups and rings makes other math feel like an engineering exercise. Making new structures and proofs redefined what I understood math to be.
If Math is a language, then the first 10 years of instruction are essentially spelling tests. There is very little flexibility nor room for creative thought in those years of instruction. Most kids never get to a proof or other areas without a defined path from start to finish. That is like only studying English grammar and never taking a literature test. Sure it can be interesting, but it is hardly creative.
I see mathematics, especially the more abstract areas, as much more similar to the arts than to the sciences.