00dIntroduction

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

If 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.

Cognitive scientist George Armitage Miller states, “The evolution of language enabled many individuals to think together. Social units could form and work together in novel ways, cooperating as if they were a single super ordinary individual.”4 One emergency worker can, for example, using mostly sounds, organize ten men to lift a heavy car off a victim; acting together, if you will, as one super human. Spoken languages give dramatic demonstration to the fact that using symbols to represent ideas can be extraordinarily powerful.

Negative Side Effects of Using Symbols

Despite being essential for the expression of mathematics, symbols are notoriously bad in that they can quite naturally mask what is happening conceptually. This unfortunate side effect of using symbols is one of the central issues that math education must overcome.

Since symbols in mathematics rarely look like what they describe (e.g., the tally marks or rocks discussed earlier look nothing like the cows they represent), using them necessitates a temporary separation between the problems and motivations and the method of solution. This in itself isn’t a problem. The problem arises when this separation is taught as being the natural state of affairs—or worse yet, the only state of affairs.

When this happens, it can become difficult for students to acquire a proper perspective of what the symbols are really doing for them, and since students are people and not machines, this has psychological implications that can prove fatal to their understanding, and forever affect their attitude toward the entire subject of mathematics.

On the other hand, for the symbols to be most effective, they must be allowed this separation (i.e., the rocks and tally marks must be allowed to represent other things besides the cows). Only when the symbols are allowed to free themselves from their origins can they really fly and open up whole new worlds to those who use them.

So we have an interesting paradox.

The very strength of mathematics, its use of efficient and unfettered symbols and procedures to express and transform ideas, is also its greatest pitfall in terms of conceptual and contextual understanding. This paradox is real and unavoidable. The situation can be likened to the difficulties with faithfully capturing in writing what someone said or thought.

Issues with Making Communication Visible

Imagine this entire chapter without any punctuation marks, spaces between words, or capitalization. The quote by Ian Stewart at the beginning of this chapter would read as:

oneofthebiggestproblemsofmathematicsistoexplaintoeveryoneelsewhatitisal laboutthetechnicaltrappingsofthesubjectitssymbolismandformalityitsbaffling terminologyitsapparentdelightinlengthycalculationsthesetendtoobscureitsreal natureamusicianwouldbehorrifiedifhisartweretobesummedupasalotoftadpoles drawnonarowoflinesbutthatsallthattheuntrainedeyecanseeinapageofsheetmusic inthesamewaythesymbolismofmathematicsismerelyitscodedformnotitssubstan ceianstewart

as opposed to

It is easy to see that punctuation marks, spaces between words, and capitalization are a tremendous help in making passages such as Stewart’s quote more convenient to read and understand. These are issues of writing not speech; when statements are spoken they come at us in a very different fashion than they do when we attempt to make them visible.5 A major task in writing then is to design it in such a way that it recaptures, as much as possible, what is being spoken or thought.

If a statement is spoken, the speaker adds tone, facial expressions, eye contact, and gestures to the statement to convey meaning. If the statement is a silent thought, then the thinker has context, images, and emotions in mind when he is thinking the thought.

In either case, when the statement is penned to paper, these extras (tone, gestures, mental images, emotions, etc.), the nonverbal cues, if you will, are lost. The writer can make an attempt to recapture these; by adding context, formatting, careful phrasing, and through sprinkling the written text with punctuation (we will call these the “writing extras”).6

If the statement is a question, then the question mark symbol can be used to communicate this. If the speaker or thinker wanted to express the statement with emphasis, then the written form can be given an exclamation point, quotation marks, boldface or italics, bullets, set in a tabular format, and so on.

The writing extras help to clarify the speaker or thinker’s intent—making them a contextual and conceptual illumination of sorts. These extras rarely if ever do this perfectly but serve as a tremendous aid in helping written communication stand on its own.

Conceptual and Contextual Illumination

We might ask a similar question of explaining mathematics: How do we present mathematical material such that the average reader, with the appropriate background, can understand in a substantial way what is truly being communicated about the mathematics, and then have that reader also gain a true appreciation of the subject from that understanding?

It is important to recognize that the written expressions of mathematics have many of the same issues that language writing has in trying to effectively convey context and meaning. Moreover, many of these issues are of a far greater intensity in mathematics due to the simple fact that unlike language writing, which is intimately connected with the spoken language of the reader, mathematical writing seems more like a foreign language.

This is an extremely serious problem in mathematics education as the celebrated author Lancelot Hogben has alleged, with historical justification: When a subject loses contact with the common man it runs the risk of becoming a superstition.7 And it is a problem that really exists to some degree at all levels of the subject. Even exceedingly competent mathematicians and scientists face this problem to some extent when trying to understand and utilize an area of mathematics with which they are not familiar.

If it is to make the wonders and beauty of mathematics more generally accessible to the public, mathematical writing, far more so than language writing, needs to be significantly enhanced through conceptual and contextual illumination. Doing this properly is an enormous undertaking and can be likened in scope, perhaps, to the difficulties in making many of the scenic marvels in the western United States, at one time reachable only by intrepid explorers, accessible to the average citizen. Accomplishing this necessitated massive construction projects involving the creation of thousands of miles of roads, hundreds of bridges, whole cities or towns, facilities, signage, and an infrastructure involving thousands of people to manage it all.

It is with an eye toward contributing, in some small part, to this massive undertaking that How Math Works was written. Being an experiment in exposition, it aims to provide in a deliberate fashion some of that critically needed conceptual and contextual illumination.

Final Words

The universe is very generous in that it allows us to gain remarkable advantages by converting or substituting one set of things into or for another set of things. A magnificent application of this is the depiction of meaningful ideas/objects whether physical or abstract in the form of coded symbols (i.e., symbols that look nothing like what they describe). It is such a natural thing to do in mathematics that symbols have indeed become the public face of the subject. The great advantage to employing them is that it is far easier and more advantageous in general to use and manipulate symbols than it is to use and manipulate the things they stand for. This advantage gives human beings the breathtaking ability to accomplish absolutely astounding feats from extremely comfortable positions.

Mathematics, however, is more than just the manipulation of symbols, and the goal here is to shed critical light on this fact. Our discussion will revolve around what have historically been considered the five fundamental operations of elementary arithmetic. These include the standard four of addition, subtraction, multiplication, and division, as well as that of the representation of quantity, often referred to as numeration. We will focus on whole numbers exclusively.

This material may seem too shallow to base an entire book around but, as you will discover, nothing could be further from the truth. There is great depth, beauty, and genius inherent in these five operations and the framework surrounding them, and we will attempt to bring awareness to this fact by flooding the conversation with conceptual and contextual illumination.

Many of the questions we will address include:

• What does it mean that our number system is “base ten”? What is the significance of the notions ten, hundred, thousand, . . . , million? (See chapter 1.)

• What do language writing and mathematical writing have in common? Can any of this be useful in illuminating issues in mathematics? (See chapter 2.)

• Where does our number system come from? (See chapter 3.)

• What is the modern significance in math education of the ancient tool called an abacus? (See chapters 3, 5, 6, and 8.)

• Why do some number systems, such as ours, have a zero and other number systems, such as Roman numerals, have none? What are our numerals called? (See chapter 3.)

• Why do we care about mathematics? (See chapter 4.)

• Where do the vertical numeral formations we use in addition, subtraction, multiplication, and division originate? (See chapter 5.)

• What is the true significance of a times table? Are there tables for the other operations? (See chapters 5, 6, 7 and 8.)

• What makes the multiplication algorithm really tick? (See chapter 7.)

• What are three major interpretations we can give to division? Why is division by zero undefined? (See chapter 8.)

• What is going on with the long division algorithm? (See chapter 9.)

• Why did the numerals we use today replace all of the others? (See chapter 10.)

• What was arithmetic education like 500 years ago? How did medieval Italian merchants and eighteenth-century Swiss educators influence elementary arithmetic education in America? (See chapter 11.)

• How do numbers help illuminate our world? What do measurement and counting have in common? (See chapter 12.)

It is now time to begin our celebration of this most fundamental of subjects. I am excited! Hopefully you are too and will enjoy the journey.