How Math Works: A Guide to Grade School Arithmetic for Parents and Teachers

305 Pages·2013·9.41 MB·English

by  G. Arnell Williams

HOW MATH WORKS

HOW MATH WORKS
A Guide to Grade School Arithmetic for Parents and
Teachers
G. Arnell Williams

ROWMAN & LITTLEFIELD PUBLISHERS, INC.
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Copyright © 2013 by G. Arnell Williams

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In dedication:

To my friends and colleagues, Joseph Mischel and Debbie Prell, who set
marvelous examples for friends and family alike by their bravery, dignity, and
positive outlook in the face of life’s greatest hardships.

To my California State University, Long Beach Professors Saleem H. Watson
and Howard J. Schwartz; both of whom, in freely sharing their insights, gave me
a passion for sharing my own.

In celebration:

Of all the men and women (known and unknown) throughout the ages who have
contributed to our understanding and use of elementary arithmetic.
Acknowledgments
T
HE WRITING OF THIS BOOK has been an exciting journey for me. To attempt to
tackle an old and familiar subject with fresh eyes has certainly been a challenge,
and throughout I have received assistance from many people on varying levels.
First, I would like to express my gratitude to Rachel Black for her patience and
support during the earliest years of this project. For their time in plowing
through those very early and very rough drafts, I say thanks to Carl Bickford and
Callie Vanderbilt. The feedback they provided gave me a better feel for how to
reach the book’s intended audience.
Those who provided useful commentary during the middle years of this
project include Carol Jonas-Morrison, Eric Bateman, Jim Phillips, Judy Palier,
Orly Hersh, Andi Penner, Karla Hackman, Russ Whiting, Pete Kinnas, and
Michelle Berkey.
The software program Adobe Illustrator proved critical throughout the entire
project, and honestly, I’m not sure I could have created the presentation I was
trying to build without it. Thanks to Fionna Harrington for suggesting the
program and for her technical assistance. Lynn Lane deserves a note of praise in
providing much needed expertise in the finer details of Microsoft Word. Thanks
to Stephen Gillen for sharing his extensive legal insights.
Barbara Waxer was gracious in offering advice on copyright issues for which I
thank her. And Wendy Vicenti deserves a special note of praise for the
invaluable feedback she gave on the text from a student’s point of view. Bill
Hatch was truly generous in lending his artistic talents in the drawing of the
Egyptian hieroglyphic numerals.
In any undertaking of this size, an encouraging word or a piece of advice from
friends or colleagues proves more useful than they may realize—especially after
rejections of book proposals start to occur. For their words of encouragement
and advice, I thank Vernon Willie, Angela Grubbs, Sylvia Hensley, Valeria
Christea, Kay Brown, Vonda Rabuck, Nancy Mischel, Sherry Nagy, Lynn
Jensen, Cheryl Palmer, Nancy Sheehan, Kathleen Chambers, Melanie
McAllister, Larry Wilson, Ken Heil, Andrea Ericksen, Greg Ericksen, Laurie
Gruel, Shelley Amator, Michelle Meeks, Jon Oberlander, Ann Meyer, Aerial
Cross, Laura Kerr, Traci Hales-Vass, Patrick Vass, Susan Grimes, Jeff Wood, and
Jenia Walter.
For his willingness to be a constant sounding board as I bounced ideas around
as well as for providing insights from his deep well of knowledge, I give
particular thanks to Sumant Krishnaswamy.
The last several months have proved the busiest in finishing this work and
several individuals provided timely information and advice at key moments. I
would like to personally convey my appreciation to Seth Abrahamson and Kate
McGowan for giving an involved read to a huge chunk of the text as it neared
the final draft stage. Their comments and thoughts proved especially valuable.
A debt of gratitude is also owed those at Rowman & Littlefield who helped
me turn my ideas and manuscript into the finished book. I am especially
indebted to Patti (Belcher) Davis for believing in me from the start and taking
the time to plow through my rough proposal to her as my acquisitions editor—
making insightful suggestions on how to improve upon it. If this book proves a
good read, it is due in no small part to Patti’s advice. A special thanks as well to
Suzanne Staszak-Silva for hearing my input and being willing to take up the
baton to help me see this project through to completion. Kathryn Knigge was
also helpful in going through my final manuscript page by page and figure by
figure, for which I thank her.
These acknowledgments would not be complete without me giving a special
note of appreciation to my family, in particular my mother, Geneva, and sister,
Jennifer, for all they have done for me throughout my life.
And finally, perhaps the greatest thanks of all goes to my students. It has been
one of my true joys in life teaching them and it is they who have provided the
crucial backdrop on which the core of the ideas in this book were developed and
have played out.

G. Arnell Williams
September 2012
Introduction
One of the biggest problems of mathematics is to explain to everyone else what it is all about. The
technical trappings of the subject, its symbolism and formality, its baffling terminology, its apparent
delight in lengthy calculations: these tend to obscure its real nature. A musician would be horrified if
his art were to be summed up as “a lot of tadpoles drawn on a row of lines”; but that’s all that the
untrained eye can see in a page of sheet music. . . . In the same way, the symbolism of mathematics is
merely its coded form, not its substance.
—Ian Stewart, British mathematician and celebrated popular math and science author1
I
F YOUR CHILD ASKED why we learn a times table for multiplication but aren’t
taught one for division, what would you say? It’s a basic question. Can you
answer it? Are you able to show your child how to do long division, but can’t
explain why it works? Not just how to perform the method, mind you, but what
really makes it go? We all use the symbols {0, 1, 2, . . . , 9} every day: Do you
know where they came from or what they are called? What do you call them,
and can you explain to someone why we calculate with them instead of with
Roman numerals? By the time you finish this book, you will know the answers
to these questions and many more, even the most important one that all parents
or teachers have been asked: Why is this stuff important?
Put succinctly, this book is for readers who want to know the why in
arithmetic—not just the how. If you want to know the context in which
arithmetic sits and where the techniques come from, then you have come to the
right place. In these pages you will find explained not just how to do
multiplication but also what actually makes it tick and how our ancestors tamed
it. If you are comfortable in your understanding of the rules of elementary
arithmetic, you may still be surprised to learn how much is really involved in
making the rules work. If, on the other hand, you are not content in your
conceptual understanding of arithmetic and desire to significantly enhance it,
then you won’t be disappointed.
You may have heard the experts wax eloquent when discussing mathematics,
describing it as powerful, mysterious in its reach, even beautiful. Are they
serious? To a supermajority of humankind these adjectives are completely
invisible when they see mathematics expressed on paper.
My hope with this text is to breathe life into some of that magic and beauty
mathematicians rave about when describing their subject. I will attempt to do
this by seizing upon them at the fountainhead, for believe it or not, the beauty
and power of mathematics are not confined to the higher realms of the subject,
but are present in elementary arithmetic right from the start. Conceptual jewels,
accessible to you, are available for the taking, and it is my intention to open
these up in conversation and view them in the brilliant light of context and
history.
While all are welcome to join us on this journey, this book is specifically
targeted to address the needs of the general adult reader who, while not being a
mathematician or scientist, is nevertheless curious about what mathematics is all
about and wants to significantly increase their conceptual understanding of the
subject. Hopefully in its reading, you will find that elementary arithmetic is truly
spectacular and thereby gain a new appreciation and understanding of the subject
in a way that allows you to better deal with the mathematics you might
encounter in your life, better explain it to your children, or better understand
other math and science books that you may read.
There Is More to Mathematics Than Symbols
A key ingredient in appreciating what mathematics is about is to realize that it is
concerned with ideas, understanding, and communication more than it is with
any specific brand of symbols. And while symbols form a crucial centerpiece in
all of this, they are not the goal in and of themselves.
In terms of using ideas in extremely powerful ways, mathematics holds an
exceptional, almost hallowed place. It is no stretch of the English language to
say that ideas and reasoning cast in mathematical form are truly something else.
The great Galileo is said to have declared that, “Man’s understanding where
mathematics can be brought to bear, rises to the level even of god’s.”2
It is almost as if ideas set in mathematical form melt and become liquid and
just as rivers can, from the most humble beginnings, flow for thousands of miles,
through the most varied topography bringing nourishment and life with them
wherever they go, so too can ideas cast in mathematical form flow far from their
original sources, along well-defined paths, electrifying and dramatically
affecting much of what they touch.
For us to dial into this transportability, however, requires that we use symbols
—a lot of them in fact (think of the symbols as part of the fluid and the rules of
mathematics as part of the riverbed). It is through the use of symbols that human
beings can leverage the almost magical ability of mathematics to systematically
and reproducibly transform ideas into other ideas, and the need for them appears
quickly when we try to answer questions involving quantity.
Why Symbols Are Needed in Arithmetic
People have always had the need and desire to compare and analyze the sizes of
collections. How much stuff do we have? How many people are in our
settlement? How large is our enemy? Collections, such as these, vary in size and
when we get to the point of describing or cataloging these variations in-depth,
we are inevitably led to symbolic descriptions. How do symbols help us? Let’s
take a peek.
Consider a scenario involving two cattle ranchers, each with a large herd
numbering into the thousands, wanting to know who has more cows. For the
time being, let’s assume that no system of numeration has been developed and
that they must figure out a way to do the comparison from scratch. How will
they be able to prove, beyond dispute, who has the larger herd?
There are several ways to proceed. One involves the ranchers creating a pair
of lanes (one for each herd) and then having their ranch hands round up the cows
and march them singly down each of their respective lanes in a matching off
process. If the herds are of unequal size, one of the ranchers will eventually run
out of cows in the pairing. The one with the excess of cows can then conclude
that he has the larger herd. While this method certainly works in determining
who has the larger herd, it could be very difficult to accomplish in practice.
There are better ways.
Another method involves using two carts (one for each herd) and a large
collection of small rocks. Each rancher’s herd is now measured by going out into
their respective pastures and placing a rock in their respective carts for each cow.
Once each herd has been measured in this fashion, it is a much simpler matter to
bring the carts in close proximity and pair off the small inanimate rocks than it is
to round up and pair off two sizeable herds of huge, living, smelly animals. The
ranchers can obtain the same information as with the first method but this time in
a much more convenient manner.
Each rock in the collection has acquired a new meaning—rather than simply
being a rock, it now stands for a cow. Or put another way, each rock has become
a symbol.
Two great strides are gained by taking this simple step. First, it is clearly much
easier and more convenient to match off small inanimate rocks than it is
matching off hundreds of large animate cows, each with its own agenda. Second,
using the rocks as symbols has opened up a vastly superior way of comparing
collections. Given that existence of an object is what counts in whether a rock is
placed into the cart, there is nothing that prevents the ranchers from comparing
other things that exist besides two herds of cattle. They could just as easily use
these carts and rocks to compare the sizes of two groups of people, two
neighborhoods of houses, two forests of tall trees, and so on. For many of these
situations, the two lanes method is impossible to use at all. Large houses or tall
trees cannot be easily rounded up, marched down lanes, and paired off. So we
see that the method with rocks is not only more handy than the method with
lanes, it also gives the ability to compare a greater variety of objects.
Since they are in the mood, can they find any symbols more convenient than
using rocks? Absolutely! If the ranchers had some sort of portable writing
system, they could replace the rocks in the carts with written tally marks. For
instance, they could use any of the following sets of marks: |, X, O, or +. If they
chose to use |, three rocks in a cart would be represented as: | | | .
Once each had done his separate tally of his respective group, the ranchers
could simply compare or match off the written symbols and no longer be
burdened by pulling heavy carts full of rocks. And since tally marks can be
created at will whereas rocks cannot, tally marks can, in theory, measure much
larger collections without as great a concern for supply issues.
Each of these improvements can be looked upon as a “technological”
breakthrough in how collections are measured, and it is clear to see that the
method of indirect comparison, in this case using symbols that stand for the
objects being counted, has decisive advantages over directly using the objects
themselves. Throughout this book, we will see that in mathematics symbols are
absolutely necessary.
Symbols Are Important in Language as Well
The need for using symbols is not unique to mathematics. Other systems
critically depend on them as well. The most familiar of these are spoken
languages. Spoken languages are systems that use sounds as symbols. They give
us the remarkable ability to describe and communicate with easy to produce
sounds as opposed to trying to do so by reconstructing, out of thin air, the
physical objects, events, and ideas that we wish to describe.
In other words, spoken languages give us the ability to represent a substantial
portion of life through the use of nothing but sounds.3 Speaking allows us to take
our inner thoughts and share them with others by simply making sounds with our
vocal chords. A song consisting of nothing but sounds can bring people to tears
or motivate them to action. Think of the organized sounds of speech serving as
part of the “fluid” for transporting thoughts and emotions just as numerical
symbols are part of the “fluid” for communicating quantitative information in
mathematics.