land_of_ice said:
One thing that can REALLY mess you up gradewise is trying to figure out how to do a problem, and spending so much time tinkering with it that you are not getting any actual homework done, you may think you've spent countless hours studying but you were not studying right so you get an F on your test or whatever the case may be.
Does anyone have tips on studying math that puts you on top, or books that actually helped you to get better grades in math by improving your studying habits ?
I'm a math major myself and through other peoples comments and work I've learned a few things that have helped me understand math better.
A couple of questions that you could ask yourself include:
What is math all about?
A lot of people will have their own opinions on this but i'll answer what I've learned math is all about. The following is a list of things that to me describe what math is all about:
a) Decomposition
In a general analytic situation the goal of trying to figure out something to get to some solution(s) involves what is known as decomposition of a problem or "breaking down". Differenty areas of math including graph theory, number theory, Fourier analysis and more describe a particular methdology of decomposing some system into its subsystems all the way down to some atomic means of measurement.
From this, its easier now to see why good mathematicians can work in areas that are undeveloped and help develop them in regards to analysis and synthesis of a particular field of science.
Once you learn and realize the many different ways of decomposing a representation so that this decomposition leads to a breakdown whereby it becomes (hopefully) simpler to analyze and make sense of, you will be able to see systems for what they really are and step back and realize what the system is all about.
Learning to decompose or "atomize" systems is not a trivial thing however. It's taken many thousands of years to come up with classification and decomposition schemes in mathematics and although a lot of schemes will seem "obvious" when you are taught them, it's not always trivial to create these systems out of nothing from the human mind alone.
b) Analysis
After decomposition, the next likely candidate is analysis. From statistics to pure math and back to more applied mathematics, analysis is an extremely important framework for analyzing general mathematical systems.
The current methodology of analysis has its roots in calculus. Calculus in a nutshell is the study of how modelling change can be used to gather various characteristics of a system.
Systems can be modeled in a variety of different ways from explicit or implicit function
definitions to differential and even partial differential equations.
If you know how a system changes with respect to the variables of that system then you
can basically model the system and analyze its characteristics.
There are so many ways of analyzing a system however although the term "Analysis" is generally associated with "Real or Complex Analysis". In statistics we use set theory and the notion of probability spaces that define a random variable to outline the framework of analyzing "random" behaviour through distributions (both univariate and multivariate).
One thing to remember with statistics is that the assumption is made is that given a distribution that we think "fits" our data, we are making the assumption about the long term probability characteristics of a specific system. This brings me to talk about another important thing about mathematics called "Assumptions" or "Axioms"
c) Assumptions or Axioms
It doesn't matter what area of mathematics we are involved in whether pure, applied, or statistical, we always are making assumptions which are used to later define some behavoural characteristic or to aid in attempting to prove something.
Once you understand how your initial assumptions translate to a model or a proof then you will be able to grasp not only the systems characteristics but the "big picture". The assumptions you make set limits for what you are trying to achieve.
In statistics we assume that if we did an infinite number of tries in a system then the distribution reflects the results that we get. In pure math we start out with axioms which are generally used to help prove a particular thereom which will build on our axioms and involve usually a set of creative mathematical transformations that help prove our claim.
One important thing to realize is that assumptions for something to be random is usually simply an easy way out of not modelling a complete system that describes the relationship between all variables completely.
After axioms start we have to deal with a unified process known as "transforming"
d) Transformations
Transformations play an extremely critical role in the method of mathematical analysis and in "recomposing" a system or axioms or theorems to eventually obtain a proof of a theorem.
An example of transformations is a Fourier transform or a laplace transform. Essentially we are taking one representation and turning into another whereby the equivalence of the two systems is such that it represents the same system or the same system to some level of approximation.
Transformations in some instances will take a system and will change it. For example finding the inverse of a system you can apply a transformation of the original linear system to get the inverse transformation of the system.
The key thing with transforms is that when you decide to transform one representation to something else, the whole system must stay "balanced" or must approximate it to some
known and provable threshold (where error of the system can be determined).
You could for example use Fourier analysis to come up with the signum function and come up with a continuous analytical function that approximates the function with very high accuracy.
So far I have tried to use some terms that describe the elements of many areas of mathematics without resorting to the actual area explicitly.
Most people will be forced to take introductory courses in most areas of mathematics including analysis algebra and possibly topology probability differential equations all calculus courses as well as complex analysis and either stats, more pure math, or more applied math.
A lot of these areas are becoming more abstract in the sense that most mathematicians will take the current research attempt to understand it and do the following:
a) Generalize
By this method, the highest level of abstraction becomes a subset of a new unifying mechanism whereby the new mechanism uses a different viewpoint or some mechanism that describes the field in terms of some key ideas. These ideas are extended to help build the specialty area up so that tools are made available that may analyze and highlight key characteristics of that particular area of study
b) Synthesize
By this I mean take the current foundation of some area and bring it together possibly with another area which may help explain a particular result or may improve the areas understanding of a particular model or area depending on the type of math involved.
c) Recreation or modification of assumptions
Through probability its assumed that a macro-level of behaviour is examinable from some system. One method where you deal with stochastic calculus is where random variables
and calculus merge together.
You may for example want to use some fancy conditional probability assumptions in your SDE that help model financial prices based on some economic assumptions about conditional dependence of the price.
When you realize and can clearly see how each assumption both singularly and collectively affects the important or sought after characteristics of the system then you will be able to take "a step back" and see what is going on.
No matter what area of science whether its economics, financial engineering, physics, chemistry, whatever, understanding the assumptions and their impact on the model is critical in being able to analyze and make sense of what you need to make sense of.
To help sharpen these abilities I recommend you also study outside mathematics and learn about structures, decomposition, analysis in different areas.
Take for instance the analysis of colour. There are probably half a dozen models for colour include HSV (Hue Saturation Value), CMYK, RGB, CIE, NCS and others.
Each decomposition method has its own advantages and disadvantages and come from studying things like vision in living things as well as specific applications in mind given a colour model.
Some books I can recommend include Polya's How to Solve It and the Princeton companion to Mathematics.
I wish you all the best
Matthew
