Struggling with long computations. Any advice?

Question

Sorry for the somewhat strange question but I have been struggling with this for some time now.

I am currently in undergraduate Electrical Engineering taking classes on Linear Algebra, Calculus and ODEs. Whenever I need to solve problems of Linear Algebra, Laplace Transforms and even computing some big integrals, I become disorganized with the long computations required (even simple ones) and I end up with the wrong answer.

I also become very anxious all of a sudden, speeding up with the computations and making silly mistakes on basic operations.

When I finally finish, I find myself with long pages of computations and a wrong answer.

How can I overcome this? How can I keep everything organized even in long problems and stay calm during the entire computation? How do you guys manage long computations like these?

Any advice is greatly appreciated!

Answer 1

This is an important methodological question.

Yes, there is inevitably a certain rate of (at least) mis-copying error, if nothing else. So we do need sanity checks (which may be easier), and, also, precision-checks, all along the way.

And, in particular, it would be extremely naive to think that "a mere computation" is a trivial thing to accomplish. It is not. Double-checks, checks on sub-cases, sanity-checks, "parity-checks", and all these things are standard operating procedure. At least if the outcome really matters!!! :)

(The idea that proof-checking software, etc., will solve such problems is a bit naive, because it depends on essentially-perfect data entry, and essentially-perfect software set-up. The infinitely-wiser version of this, that can guess what "we meant to say" is not quite here... :)

Answer 2

I don't know. There's two aspects when it comes to proving things in Linear Algebra, Calculus etc. a) The proof sketch and b) The proof as a logical sequence of statements. I think being good at the "proof sketch" comes after you've spent lots of time studying the subject. So it takes experience to get good at. For b), The formal logical sequence of statements part, I simply don't allow myself to write anything that is false. If I'm not sure if something is true or false, and I want to determine whether or not something is true, I ask myself whether or not that statement is true as a question ("Is Statement X true or false?"), and then try to figure out whether or not it's true. Because I have invested time doing this, the result as to whether or not Statement X was True, sticks in my memory. I don't just pretend that it's true and roll with it forever, because then you won't remember what's true and what's not.

This approach can help you not accept commonly used false statements like Freshman's Dream, which a lot of students I have tutored use in their working when they're "just trying to get to the answer". You won't get to the answer with false statements. People don't want to spend any serious amount of time doing problems. However, the only way to improve in maths is to study problems and solve them rigorously.

Improving your maths takes serious time and effort, and there's no way round this.