Arithmetic Magic - The Introduction
So you think you're not good with math. Hmmm. Well that's
something I hear all the time so you shouldn't feel too bad. The
truth is, many, many people, at one time or another, have felt
the same way, and have experienced something which has made them
feel bad about math and numbers in general. So the common
complaint, "I was never good at math," or "I hate numbers and
math," is as common as the common cold and sometimes just as
annoying. Interestingly, what I have discovered from working
with students through the years is that this feeling of
inadequacy often stems from some negative experiences with the
subject. This might very well be the case with you. Perhaps you
had a hard time in one of your math classes and felt very
frustrated by it. Perhaps you struggled year after year because
you could'nt understand the material being taught, or because
the pace was too fast. Perhaps you didn't understand because the
subject matter was presented in a way that did not get across to
you. Sound familiar?
My concern with the above scenario is that, uncorrected, this
problem usually leads to feelings of frustration and inadequacy
toward math in general. A likely--yet very undesirable--end
result would be the drawing of incorrect conclusions about
yourself and your abilities. Do these sound familiar?: "I'm just
not good at math and never will be," or "I'm just not smart
enough to do math," or more general and even more crippling,
"I'm just not that smart." I cannot count how many times I have
heard such statements, the like of which only serve to drain the
energy and stunt the educational gains of even the best of us.
Unfortunately, these statements crystallize into attitudes and
beliefs which become difficult to shake off. However mistaken
these beliefs might be, their burdensome influence remains for a
very long time-- sometimes even for life. The end result--math
illiteracy.
This is very unfortunate--even tragic--since the ability to work
with numbers and do math are skills that have countless
benefits. After all, no one should be doomed to mathematical
illiteracy, especially when the alternative is so near within
reach. Let's consider the importance of basic math skills in
everyday life. For one, you couldn't count without math, pay
bills and then balance a checkbook, understand basic financial
matters, such as by how much money will grow at different
interest rates, or even figure how much change you should get
after a simple purchase transaction. What a sorry state we would
all be in if we couldn't do these simple things! Now think for a
second. Suppose you could add, subtract, multiply and divide
numbers so that routine calculations were no longer a bother.
Numbers are now no longer your enemies but your friends. A very
likely consequence would then be that the word math no longer
triggers a bunch of negative impressions but rather positive
ones. You now consider yourself good at a very tough subject and
therefore consider yourself a "cut above the rest." The
implications of this new self-appraisal are enormous.
Furthermore, imagine having a simple method which allowed you to
figure percents easily. Then you could figure discounts in a
store and know exactly how much less you would pay for items
that go on sale. As you will soon see, working with calculations
involving percents is nothing more than understanding decimals
and how to multiply numbers. Of course, rather than figure the
discounts yourself you could wait until you got to the cash
register to pay for the items. This would always be a
possibility, but once again these methods do not only have one
practical use. If they did, their value would be that much less.
Percents figure in many other areas of the real world, not just
in purchase transactions. In short, these newly acquired skills,
whether they be in the area of multiplication and addition, or
in the area of percent problems, will certainly open up all
sorts of possibilities within the mathematical realm and unleash
capabilities within you that you never thought you possessed. In
short, this is the aim of Arithmetic Magic: to foster an
appreciation of mathematics through the study of the basic
arithmetical operations.
With this goal in mind, many positive things are achieved.
Specifically, as you learn the techniques in this book, your new
ability to work with numbers will definitely give you an
advantage--an edge, sharp and cutting--to whatever you do. Aside
from giving you a command over numbers, which will come in handy
in daily life and serve you countless times in your days going
forward, the techniques will pave the road toward new-found
confidence and a boost in attitude towards your mental
abilities. This--more than the computational ability--is the key
payoff. Personally, it would be difficult for me to estimate the
number of times I have used these techniques or the number of
ways in which they serve me. In fact, I am constantly coming up
with new combinations of the techniques and teaching both myself
and my students new methods based on existing ones. You will
find yourself doing the same as you read this book with an open
and hungry mind.
For those of you who think the ability to work easily with
numbers is beyond them, think again. From my experience, I am
convinced that the ability to work well with numbers is not
solely a byproduct of inborn talent. This is in spite of what
many might think. Barring any severe mental abnormalities, an
individual can improve his ability at math. Even the so-called
"classified" individuals, whether they be considered "learning
disabled" or afflicted with ADD (Attention Deficit
Disorder)--whatever such classifications might mean--have
benefitted from these techniques. Part of the inability to do
math seems to be simply that, when it comes to math and working
with numbers, many individuals concede defeat before even
trying. Because such people "throw in the towel" before the
fight is over and never attempt to do the math, they reach the
mistaken conclusion that math and numbers are just not for them.
An erroneous, yet very sorry, conclusion.
Consequently, if you've been frustrated by math-- and its
fiendish counterparts, numbers--it should be refreshing to hear
what I have discovered regarding this particular illness: you
can do math and you can work with numbers. This is something I
am sure of and, by the time you've read this book, something I
will have proven to you. I can state this positively from
experience. What you need, and what is absolutely essential, is
the right approach. Time and time again, I have found that many
people cannot work with numbers because they have simply
convinced themselves that they are bad at math. Why? It seems
that these unfortunate individuals experienced some frustration
with the subject early on. Because these early failures have far
outweighed any successes, these same individuals come under the
mistaken notion that they cannot do math. This situation now
sets in motion a negative cycle. These early failures--or "lack
of successes"--snowball to the point of no return and form a
downward spiral. Once this happens, the individual is then
certain that math is not a fun subject and one to be avoided at
all costs. After all, you can't do something if you believe you
can't.
Admittedly, mathematics is not an easy subject. Math has its
roots in numbers and although the more abstract branches of this
subject might find themselves far removed from numbers, the
foundation in numbers is still there. You cannot escape them.
There is even a whole branch of mathematics devoted to the study
of numbers and number properties called Number Theory. It would
seem then that the royal road to understanding math lies in
understanding numbers and that the two were inseparable. I say
royal because even such famous mathematicians as Pythagoras--you
know the one with that famous theorem in geometry-- have written
that numbers and counting formed the foundation of all
mathematics. Moreover, since math--with its many branches and
applications-- serves as one of the principle tools to
explaining the world around us, and since these things we call
numbers find themselves intricately woven into all of
mathematics, a good understanding of them can certainly help
pave the way to a better understanding of mathematics in
general. Even doing calculus requires that you understand the
basic arithmetic properties of numbers. As though this were not
all, one very real and definite benefit in learning and studying
mathematics comes from the educational gain made in rational and
logical thinking. Now that's a nice payoff!
What I have laid out here in four simple chapters is the basis
for understanding numbers. This is done through a study of the
four basic arithmetic operations: addition, subtraction,
multiplication, and division. As you will soon see, each of
these operations finds itself related by special properties to
the other. This relationship allows us to simplify even further
the study of these basic operations. I do use some basic math
terminology but I deliberately avoid getting weighted down in
special names. Far more important than memorizing any
terminology is understanding and learning the techniques and
knowing how and when to utilize the methods.
As someone who struggled with math--who understands the
frustrations and feelings of inadequacy that this subject can
rain upon you--I put this book together with the confident hope
that it will help you gain a whole new perspective on math and
numbers. This new attitude might foster a desire for further
study, and this would ultimately lead to a higher rate of math
proficiency. I am delighted to tell you that in overcoming my
frustrations toward math and numbers, I have benefited
enormously from gains made in selfconfidence towards this
subject and many others. I now approach any subject matter or
challenge-- whether personal or academic--with a feeling of "I
can do it." If after reading this, your attitude towards math
and numbers has changed to a more positive and less fearful one,
then my goal will have been accomplished.
One final note that should set you in a very positive frame of
mind before reading is the following: the arithmetic techniques
presented here are not only empowering but also mind
stimulating. They are confidence boosting. Now I am quite aware
that in the high technology world of today, calculators are
readily available and found everywhere; even cellular phones
have these built-in devices ready to serve you. But the truth of
the matter is that even though technology is everywhere, the use
of such technology is sometimes limited. The point of this
discussion is that there is something special about knowing how
to do something, and this is particularly true when you can do
something that others think is hard. Such as mathematics. The
other point is that knowledge in and of itself has a very
rewarding aspect to it, one that often cannot be thoroughly
evaluated or appraised. So you are not learning these techniques
solely to do away with your calculator, as this device will
always have its uses. You are learning them so that you come to
understand that what you thought was undo-able is actually
do-able and what you thought might be impossible is actually
possible. This fact is very empowering indeed and can serve as
the very bridge to higher achievement.
So let's get going and begin our journey into Arithmetic
Magic. May you never be the same again.