How to Succeed in Mathematics

Understand Underlying Concepts

Why Should I Bother?

It isn't good enough to know that $\frac{d}{dx}5x^3 = 15x^2$ simply because of the power rule.

In addition to the knowledge that the power rule is being utilized here, you also must understand why the power rule works. You need to understand the proof of the power rule; the inner workings of the proof itself should be visualized inside your head.

But why? Why should I waste hours upon hours trying to understand the inner workings of theorems and rules when mathematicians have already done that for me?
Because then you'll actually understand what the hell is going on. Without understanding all of the mathematics from the lowest level, you aren't able to visualize the inner workings of the mathematics that you are processing.

With an intimate understanding of the inner workings of the mathematics you're working with, it become much easier to think in terms of "Hey, I can solve this exercise logically, using my intimate knowledge of the interrelationships between the different facets of this problem."

Without that intimate understanding: "Hey, I can solve this exercise. Wait. Oh yeah, now I remember. I'm supposed to do this, and this, then this. Oh yeah. Ok, I think that's correct."
You will rely on memorizing those steps to solve the exercise, instead of the logic behind the math.

Proofs in the Textbook

Read the textbook very carefully, by looking at all example exercises and proofs. Your math textbook explains every rule in an extremely in-depth manner. In addition to the actual syntax and application of the rule, your textbook will go over the proof of the rule, which will help you understand the building blocks of the math involved.

Here is an example from the Larson-Edwards Calculus book, which I used: image

Here are the steps I took when I learned the underlying mathematics of the power rule:

1. Skim the binomial rule at the top. Whatever. I kind of get it. NEXT.
2. Look at the definition for the power rule. Cool beans. Now I know the definition.
3. Look at the proof. Holy crap, the second step in the proof uses the binomial rule. I guess I have to go back up to the top of the page and carefully reread the part about the binomial rule.
4. Learn the underlying mathematics of the binomial theorem.
   1. Carefully reread the content in the textbook that talks about the binomial rule. I may have memorized it at this point, but I can't yet visualize in my head the inner workings of it. I still don't understand how it works.
   2. Next, you should search the web for an explanation of how the binomial theorem works. Usually, they go through the proof, step by step.
   3. However, I did the previous step somewhat differently. I basically sat down with a blank sheet of paper, and wrote out the formula for binomial expansion. Then I started writing stuff down and working things in an attempt to be able to visually understand it. I will not explain to you my befuddled and very unique thought process, because it is MY thought process. Every person has a different way of understanding and visualizing things. In reality, if spend a few hours, I could construct a good representation of my thoughts for you to see, but it would be very difficult, and I don't have that kind of time right now. Maybe another time.
5. Integrate what I learned about binomial expansion into the proof for the power rule. At this point, I am looking at the proof in the textbook right now, and trying to understand the process of each step by understanding and visualizing all of the arithmetic operations going on.
6. Whala, I understand! Now comes a huge rush of pure deliciousness into my brain; it's the feeling that the last several hours of my life I thought I was wasting - well, they weren't wasted, because now I fully understand every single math operation that takes place in $\frac{d}{dx}5x^3 = 15x^2.$ It's a beautiful thing to experience.

Meet Privately with your Teacher

Don't schedule a private meeting with your teacher until you have read the textbook. For example, say you are extremely confused as to how to solve a a related rates problem.

• Do not - go to your teacher and say "I don't get related rates. Can you help me understand?" If you do this, you are being vague, and sending this message to your teacher: "I'm too lazy to figure it out myself."

• Do - look in your text and try to read up on what you don't understand. Try as hard as you can to learn the concept on your own. Then, take a sheet of paper and jot down a list of the things that you understand, and a list of things you don't understand. You need to have a question ready to ask, and you need to be able to explain what you already know. If you do this, then the teacher will be able to give you an explanation tailored to your misunderstanding. In other words, you will better be able to understand the teacher's explanation, because she knows specifically what you don't understand, and what to address. You are sending this message to the teacher: "I'm motivated to learn on my own, and to clarify anything that I can't fully grasp by coming to you."

Browse the Web

The internet has websites that

1. Analyze and solve equations for you
2. Give you strategies and methodologies to follow for being successful at math (like this wiki)
3. Comprehensive sites, almost like online textbooks, that go over every single aspect of mathematics that you can think of

WolframAlpha

Solve for virtually any math equation.

For example, analyzing $y = 2x^4 - 53x^2$ is quite simple: example

WolframAlpha can be used to do almost anything. For example:

1. Start by going to the home page. home
2. Click on examples.
3. You will see many topics, ranging from weather to transportation to, you named it, mathematics. Click on mathematics.
4. You will see many mathematics related functions. You probably want Calculus and Analysis.
5. Look at everything it can do. Be amazed.

Calculus Wikibook

Like an online textbook. A wik dedicated to Calculus. Contains good explanations and examples.

Practice

Do all the Homework

Ask your teacher about your problem solving method

Use the exercises in the Textbook

Collaborate

Start a Facebook Study Group

Skype

Meet Up Outside of Class

Get Ahead

(read the textbook before you come into class)

Be Neat

Being neat means you will almost never make careless mistakes. I am extremely neat when I am doing math, and I rarely make careless mistakes. When I do make a mistake, it means I don’t understand the math. However, when I start writing English, I switch into English writing mode and I become sloppy.

Use the Internet

WolframAlpha
Google