andryd9 said:
I know my post is vague, but perhaps someone can help me. I have finished my college Calculus series and will take Differential Equations next. I have done well in all my courses, but still feel like I'm not really grounded mathematically. Perhaps a Physics course, to demonstrate more applications? I asked a friend in Grad. School, who said that math as a whole starts to make more sense once you take an analysis course. A physicist I know said that some Statistics courses would help. What do you think?
Hey andryd9 and welcome to the forums.
In terms of what is going on, this is different for each field, but I'll give you my insight of what I have learned.
Mathematics to me is all about three things: representation, transformation, and assumptions.
First we figure out how to represent something. Second we make assumptions which automatically force constraints that are good enough so that we can deal with the problem we are working on at hand. Then we transform what we have into something else. We usually do the above until we get what we are looking for.
Now this is not an attempt to paint every area of mathematics with the same brush, but it does give a bit of insight into what does go on.
But if you wanted to say understand calculus on one level, you could say that it allows to calculate measures that are non-linear (think curves instead of straight lines). In high school we calculated lengths, areas and volumes out of things that had straight lines. There were a few exceptions but for the majority, it dealt with straight lines.
Calculus allows us to do this with curves, or non-linear objects. The key to understanding what the integral and differential/derivative mean is to find out exactly what parameters are changing with respect to another, and based on that you are doing what you did in high school, but you are doing it for objects that are way more complicated, and you also are looking at calculating measures more advanced than you did in high school like doing line integrals or other vector calculus problems.
In statistics, the thinking is again very different to non-statistical fields although it does share its similarities with mathematics in general. In one context, you want to find techniques to reduce bias. In another you want to find techniques that gaurantee the lowest confidence interval for the inference of a parameter. In another context you want to make sure that a theorem used to simulate from non-analytic density functions will always work (i.e. always converge to the right distribution).
The best advice I can give you to understand math is when you are in a class, ask your lecturer/TA/presenter/professor what it's all about. They have for the most part already spend a lot of time being exposed to math and most likely they have been thinking about it themselves. Don't underestimate this resource by any means.
