\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Article - How Memorizing Can Enhance Learning

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There is a lot of strong opposition to memorization when it comes to learning and, especially with math. In fact, opposition to memorization has revolutionized how math is taught in school. For many years now, there has been a trend moving toward understanding rather than memorizing. As a college student, you need to make your own choices about how you learn. It is your responsibility to take charge of your learning. On this page, we will give you our thoughts on how memorizing can enhance your mind and your learning and some books that we recommend to help you memorize.

Why We Should Still Memorize

While it is true that memorizing is not learning, memorizing is an important component of learning. Just like sitting in a classroom is not learning, but you still need to actually sit in a classroom to be in a position to learn. This is just like memorizing. We need the material in our heads to be able to learn it and use it. Memorizing gets it into our brains and trains our brain for future learning.

1. We need to be able to quickly retrieve facts and truths in order to understand them.
Just knowing a fact does not mean we have learned something. However, the important bridge between not knowing something and learning it involves memorizing. For example, we can memorize that the commutative property in multiplication means that we can multiply two numbers in either order and get the same answer. However, not until we actually notice that \(2 \times 3 = 3 \times 2\), can we say that we understand the commutative property. But without first memorizing the commutative property, we would not know what to call this operation or why it is important.
Another example, if we did not know how the letters of the alphabet are written, we would not know how to read. The alphabet and the letters must be memorized in order to be able to read.

2. Memorizing changes the structure of your brain.
We believe that this is the most important reason to memorize things on a regular basis, even when not in school. What you focus on, memorize, think about and study changes the way your brain works, making it easier to remember, focus, process and learn. Yes, in our opinion, memorizing makes it easier to focus and improves learning.

Take a few minutes of downtime to strenghten your memory. [credit: Getty Images]

Cement What You Memorized

After actively memorizing, your mind needs time to make connections and cement what you have memorized/learned. To do this, take a few minutes, 10-15, to relax and clear your mind using meditation to allow this to happen. Here is an article explaining this in more detail.[link]

Memorization Techniques

As mentioned above, there are specific techniques for memorization that make good use of your time and help you learn. However, initially, it takes some time to learn these techniques but your time will be well-spent. Here are just a few techniques.

1. Memory Palace or Method of Loci
This is the most effective technique there is to memorize most things. The idea is to build a story around a location using action and objects. Often a character is used too. The idea is that your brain remembers things more easily using a story and having several different connections. Recollection can then be done through one of several routes which makes it easier to access. The book Moonwalking with Einstein: The Art and Science of Remembering Everything is an excellent book on this technique and it is entertaining to read.

2. Number Association
A technique to remember numbers is to associate a character, an action and an object with each number (usually starting with 0-99) and then building a story to help remember the number.

Getting Started

We have only touched the surface of memorization and there are lots of other techniques. Personal experience is the only way to convince yourself that memorizing will help you learn. So you need to research this subject yourself.
To get started, we recommend that you read the book Moonwalking with Einstein: The Art and Science of Remembering Everything. This book gives you the motivation and understanding why memorizing is important and it is quite entertaining. However it lacks specific guidelines on how to implement memorization techniques.

The two books we recommend that give you specific guidelines to learn memorization techniques are
How to Develop a Brilliant Memory Week by Week: 50 Proven Ways to Enhance Your Memory Skills and
You Can Have an Amazing Memory: Learn Life-Changing Techniques and Tips from the Memory Maestro. These two books are written by the same guy and there may be some overlap, but overall they give concrete techniques to learn how to improve your memory.

We also recommend Tim Moser's site Master of Memory. He has a free course that demonstrates and helps you learn the memory palace technique. He also discusses how to remember names and faces, numbers, poems, foreign language vocabulary and other things.

This won't be easy but it will be worth it.

You CAN Ace Calculus

Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\)

\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\)

\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\)

\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\)

Set 2 - squared identities

\( \sin^2t + \cos^2t = 1\)

\( 1 + \tan^2t = \sec^2t\)

\( 1 + \cot^2t = \csc^2t\)

Set 3 - double-angle formulas

\( \sin(2t) = 2\sin(t)\cos(t)\)

\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\)

Set 4 - half-angle formulas

\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\)

\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\)

Trig Derivatives

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)


\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)

\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)


\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)

\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\)


\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\)

Inverse Trig Derivatives

\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\)


\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\)

\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\)


\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\)

\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\)


\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

Trig Integrals

\(\int{\sin(x)~dx} = -\cos(x)+C\)


\(\int{\cos(x)~dx} = \sin(x)+C\)

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)


\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\)


\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\)

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