HOW TO UNDERSTAND MATHEMATICS FORMULAS ----- BY. MWL. JAPHET MASATU.

HOW    TO  UNDERSTAND  MATHEMATICS   FORMULAS. INTRODUCTION.

In a recent IntMath Poll, many readers reported that they find math difficult because they have trouble learning math formulas and an almost equal number have trouble understanding math formulas.
I wrote some tips on learning math formulas here: How to learn math formulas.
Now for some suggestions on how to understand math formulas. These should be read together with the “learning” tips because they are closely related.
a. Understanding math is like understanding a foreign language: Say you are a native English speaker and you come across a Japanese newspaper for the first time. All the squiggles look very strange and you find you don’t understand anything.
If you want to learn to read Japanese, you need to learn new symbols, new words and new grammar. You will only start to understand Japanese newspapers (or manga comics ^_^) once you have committed to memory a few hundred symbols & several hundred words, and you have a reasonable understanding of Japanese grammar.
When it comes to math, you also need to learn new symbols (like π, θ, Σ), new words (math formulas & math terms like “function” and “derivative”) and new grammar (writing equations in a logical and consistent manner).
So before you can understand math formulas you need to learn what each of the symbols are and what they mean (including the letters). You also need to concentrate on the new vocabulary (look it up in a math dictionary for a second opinion). Also take note of the “math grammar” — the way that it is written and how one step follows another.
A little bit of effort on learning the basics will produce huge benefits.
b. Learn the formulas you already understand: All math requires earlier math. That is, all the new things you are learning now depend on what you learned last week, last semester, last year and all the way back to the numbers you learned as a little kid.
If you learn formulas as you go, it will help you to understand what’s going on in the new stuff you are studying. You will better recognize formulas, especially when the letters or the notation are changed in small ways.
Don’t always rely on formula sheets. Commit as many formulas as you can to memory — you’ll be amazed how much more confident you become and how much better you’ll understand each new concept.
c. Always learn what the formula will give you and the conditions: I notice that a lot of students write the quadratic formula as

But this is NOT the quadratic formula! Well, it’s not the whole story. A lot of important stuff is missing — the bits which help you to understand it and apply it. We need to have all of the following when writing the quadratic formula:

The solution for the quadratic equation
ax² + bx + c = 0
is given by

A lot of students miss out the “x =” and have no idea what the formula is doing for them. Also, if you miss out the following bit, you won’t know how and when to apply the formula:
ax² + bx + c = 0
Learning the full situation (the complete formula and its conditions) is vital for understanding.
d. Keep a chart of the formulas you need to know: Repetition is key to learning. If the only time you see your math formulas is when you open your textbook, there is a good chance they will be unfamiliar and you will need to start from scratch each time.
Write the formulas down and write them often. Use Post-It notes or a big piece of paper and put the formulas around your bedroom, the kitchen and the bathroom. Include the conditions for each formula and a description (in words, or a graph, or a picture).
The more familiar they are, the more chance you will recognize them and the better you will understand them as you are using them.
e. Math is often written in different ways — but with the same meaning: A lot of confusion occurs in math because of the way it is written. It often happens that you think you know and understand a formula and then you’ll see it written in another way — and panic.
A simple example is the fraction “half”. It can be written as 1/2, and also diagonally, as ½ and in a vertical arrangement like a normal fraction. We can even have it as a ratio, where the ratio of the 2 (equal) parts would be written 1:1.
Another example where the same concept can be written in different ways is angles, which can be written as capital letters (A), or maybe in the form ∠BAC, as Greek letters (like θ) or as lower case letters (x). When you are familiar with all the different ways of writing formulas and concepts, you will be able to understand them better.
Every time your teacher starts a new topic, take particular note of the way the formula is presented and the alternatives that are possible.

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Do you have any tips to add? How do you figure out your math formulas? Which formulas are hardest to understand?
Good luck with understanding math formulas!

How to learn math formulas

In a recent IntMath Poll, readers indicated that the hardest thing about math was learning the formulas.
Here are 10 things you can do to improve your memory for math formulas.

1. Read ahead

Read over tomorrow’s math lesson today. Get a general idea about the new formulas in advance, before your teacher covers them in class.
As you read ahead, you will recognize some of it, and other parts will be brand new. That’s OK – when your teacher is explaining them you already have a “hook” to hang this new knowledge on and it will make more sense — and it will be easier to memorize the formulas later.
This technique also gives you an overview of the diagrams, graphs and vocabulary in the new section. Look up any new words in a dictionary so you reduce this stumbling block in class.
This step may only take 15 minutes or so before each class, but will make a huge difference to your understanding of the math you are studying.
I always used to read ahead when I was a student and I would be calm in class while all my friends were stressed out and confused about the new topic.

2. Meaning

All of us find it very difficult to learn meaningless lists of words, letters or numbers. Our brain cannot see the connections between the words and so they are quickly forgotten.
Don’t just try to learn formulas by themselves — it’s just like learning that meaningless list.
When you need to learn formulas, also learn the conditions for each formula (it might be something like “if x > 0″).
Also draw a relevant diagram or graph each time you write the formula (it might be a parabola, or perhaps a circle). You will begin to associate the picture with the formula and then later when you need to recall that formula, the associated image will help you to remember it (and its meaning, and its conditions).
During exams, many of my students would try to answer a question with the wrong formula! I could see that they successfully learned the formula, but they had no idea how to apply it. Diagrams, graphs and pictures always help.
Most of us find it difficult to learn things in a vacuum, so make sure you learn the formulas in their right context.
When you create your summary list of formulas, include conditions and relevant pictures, graphs and diagrams.

3. Practice

You know, math teachers don’t give you homework because they are nasty creatures. They do it because they know repetition is a very important aspect of learning. If you practice a new skill, the connections between neurons in your brain are strengthened. But if you don’t practice, then the weak bonds are broken.
If you try to learn formulas without doing the practice first, then you are just making it more difficult for yourself.

4. Keep a list of symbols

Most math formulas involve some Greek letters, or perhaps some strange symbols like ^ or perhaps a letter with a bar over the top.
When we learn a foreign language, it’s good to keep a list of the new vocabulary as we come across it. As it gets more complicated, we can go back to the list to remind us of the words we learned recently but are hazy about. Learning mathematics symbols should be like this, too.
Keep a list of symbols and paste them up somewhere in your room, so that you can update it easily and can refer to it when needed. Write out the symbol in words, for example: ∑ is “sum”; ∫ is the “integration” symbol and Φ is “capital phi”, the Greek letter.
Just like when learning whole formulas, include a small diagram or graph to remind you of where each symbol came from.
Another way of keeping your list is via flash cards. Make use of dead time on the bus and learn a few formulas each day.

5. Absorb the formulas via different channels

I’ve already talked about writing and visual aids for learning formulas. Also process and learn each one by hearing it and speaking it.
An example here is the formula for the derivative of a fraction involving x terms on the top and bottom (known as the “Quotient Rule”). Then in words, the derivative is:
dy/dx = bottom times derivative of top minus top times derivative of bottom all over bottom squared.
The formula is actually as follows, if we let u = numerator and v = denominator of the fraction, then:

6. Use memory techniques

Most people are capable of learning lists of unrelated numbers or words, as long as they use the right techniques. Such techniques can be applied to the learning of formulas as well.
One of these techniques is to create a story around the thing you need to learn. The crazier the story, the better it is because it is easier to remember. If the story is set in some striking physical location, it also helps to remember it later.

7. Know why

In many examinations, they give you a math formula sheet so why do you still need to learn formulas? As mentioned earlier, if students don’t know what they are doing, they will choose a formula randomly, plug in the values and hope for the best. This usually has bad outcomes and zero marks.
I encourage you to learn the formulas, even if they are given to you in the exam. The process of learning the conditions for how to use the formula and the associated graphs or diagrams, means that you are more likely to use the correct formula and use it correctly when answering the question. This is also good for future learning, because you have a much better grasp of the basics.

8. Sleep on it

Don’t under-estimate the importance of sleep when it comes to remembering things. Deep sleep is a phase during the night where we process what we thought about during the day and this is when more permanent memories are laid down. During REM (rapid eye movement) sleep, we rehearse the new skills and consolidate them.
Avoid cramming your math formulas the night before an exam until late. Have a plan for what you are going to learn and spread it out so that it is not overwhelming.

9. Healthy body, efficient brain

The healthier you are, the less you need to worry about sickness distracting from your learning. Spend time exercising and getting the oxygen flowing in your brain. This is essential for learning.

10. Remove distractions

This one is a problem for those of us that love being on the Internet, or listening to music, or talking to our friends. There are just so many things that distract us from learning what we need to learn.
Turn off all those distractions for a set time each day. You won’t die without them. Concentrate on the formulas you need to learn and use all the above techniques.
When you are done, reward yourself with some media time — but only after you have really accomplished something.