Why does computation suck so much?

[Kindayr]

I'm in 3rd year math, and out of the 11 courses I'm taking this year, 2 are computationally based (Calc III and Calc IV).

I just finished my CalcIII midterm, I know i did fine, definitely a 90 or higher, but I just want to discuss how much I hate computation, and if anyone agrees with me.

I'm so used to pure mathematics courses where I get to actually think, and feel productive. Computation-based exams feel too overburdened with second guessing, messy objects and just plain unsatisfaction whenever I complete them. Like, even though I know I got 90+, I don't feel good about it because I didn't take anything away from it. I just showed I can plug and crank a few equations.

This differs completely to other course exams like say Topology or Group Theory, where even though I know I might only get 85, I feel like I completed something; there's a sense of satisfaction and glory.

I just wish my school offered a calculus for Math students that had different emphasis than what is in what is being offered currently.

(for those wondering, I had a double major last year and put off taking calculus until this year. now I'm a full-time pure math student, and unfortunately have it get it out of the way)

What are your thoughts?

Are computation-based courses a necessary evil for pure math students? Do you hate them as much as i do?

 

[lurflurf]

It is not computation, it is those particular computations are quite dull and included as a service to other subjects. Topology and group theory have there share of computations. Proofs are a type of computation. What is interesting about computation is determining which computations can be solved with a certain method, what method should be used for a computation, how hard each computation is, comparing general and special methods and so on. The introduction to calculus problems you mention are (at least sometimes) quite dull because they are quite simple and straight forward, the main difficulty is avoiding small errors caused by time constraints and boredom. You complain about second guessing and messy objects, but part of why they are dull is they are not messy and not ambiguous. I suppose tests could just ask write down how you would solve each problem and how well you would do, but some students might be dishonest or mistaken in there answers and it would be difficult to know. There is the opposite problem as well in some coursed the student can state and prove all the theorems, but are unable to understand or apply any of them. Some simple calculations are always a good check of understanding.

 

[homeomorphic]

I used to hate computations, and to some extent, I still do, but sometimes, I actually kind of miss doing computations, oddly enough. I think they have their place. And, although I sometimes suffer greatly at their hands, it's probably a good thing that some mathematicians actually like to compute things. Leave it to them and the computers to do the dirty work. As long as I don't have to take any PDE courses from them or anything. That's when the resentment begins (not to mention any names, here).

One thing that can sometimes be annoying is that in many theories, only the simplest examples can actually be computed without things getting ridiculous. I really don't want to compute the curvature of any surface besides a sphere or maybe the helix or some silly examples like that. Another example is Jordan forms of matrices. If it's a big matrix, there's not much point.

On the other hand, sometimes, as I noted, you can save your self a lot of work by being clever, and that can make some calculations kind of cool, and elegantly executed.

One thing that I find magical about math is when theory is combined with computations. That is what is so amazing about linear algebra. I have all these pictures in my mind about vector spaces and linear transformations and they predict the results of calculations. For example, matrix multiplication is associative because function composition is associative and matrices can be seen as representing functions. But if you try to prove it directly, it's a bit of a mess. There's something satisfying about doing a calculation every once in a while, just to see that happen.

I don't know that making calculations more difficult, in the sense of requiring more ingenuity really makes them much more interesting. Better sport perhaps, but a cheap trick that solves some integral isn't the most memorable thing. It doesn't provide conceptual understanding. Gives you nothing to take away.

 

[fleazo]

I know where you are coming from OP - but computation is like the guilty pleasure of mathematics. It's just a fun thing to sit and do even as a time waste. A fun game you know?


I agree with the second poster when he/she says that writing a proof is a form of computation. You are just taking a well defined process and applying it to arrive at some end result. But we know what you are talking about.


But just remember, that computation will keep your skills sharp. And it is also going to be useful one day if you plan to pursue research or do anything applied ever (and even the profs I admire that are pure mathematicians lend their hand to the applied side of things). You NEED to know how to take various types of integrals, derivatives, etc. Yes, it can be dull and just regurgitated, but it's useful, and hey, it's FUN! Not everything has to be mind boggling and Earth shattering. Sometimes it's just good to relax at the end of the day, sit back, and evaluate a few line integrals! :D And think about this - if you KNOW the theory, and you know it well, the computation shouldn't be any problem for you. You'll find tons of theory even in a regular old calculus class. Is it rigorous like analysis or topology? No. But there are still concepts to learn, a lot of them! Want to expand your mind and give yourself a challenge? Try and come up with different ways of solving a problem. My prof in complex analysis always says: don't solve this problem just one way: solve it the algebraic way then solve it the geometric way. Look for different routes to solve the problem. Do that in calculus - look at the problem geometrically before you solve compute it. Think about it, mull it over, learn something from it!