Book Description
We hear all the time how American children are falling behind their global peers in various basic subjects, but particularly in math. Is it our fear of math that constrains us?
Or our inability to understand math’s place in relation to our everyday lives? How can we help our children better understand the basics of arithmetic if we’re not really sure we understand them ourselves?
Here, G. Arnell Williams helps parents and teachers explore the world of math that their elementary school children are learning. Taking readers on a tour of the history of arithmetic, and its growth into the subject we know it to be today, Williams explores the beauty and relevance of mathematics by focusing on the great conceptual depth and genius already inherent in the elementary mathematics familiar to us all, and by connecting it to other well-known areas such as language and the conceptual aspects of everyday life.
The result is a book that will help you to better explain mathematics to your children. For those already well versed in these areas, the book offers a tour of the great conceptual and historical facts and assumptions that most simply take for granted.
If you are someone who has always struggled with mathematics either because you couldn’t do it or because you never really understood why the rules are the way they are, if you were irritated with the way it was taught to you with the emphasis being only on learning the rules and “recipes” by rote as opposed to obtaining a good conceptual understanding, then How Math Works is for you!
Passages from How Math Works
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. pp. xii - xiii.
A goal of this book has been to tear down in some small part these barriers to understanding by attempting to shatter the “divinity of arithmetic,” through showing that even the methods, which we now take most for granted, were not given to us from on high, but were actually the result of centuries of scientific efforts on the part of our predecessors. p. 269.
Through the judicious employment of symbols, diagrams, and calculations, mathematics enables us to acquire significant facts about extremely significant things (universal laws, even), not by first forging out into the cosmos with teams of scientists, but rather from the comforts and confines of coffee tables in our living rooms! p. 72.
In mathematics, by placing our fingers on a given problem, no matter how trite or pedestrian it apparently seems, we may end up measuring the pulse of the universe. p. 119.
Awareness of the fact that one thing in nature can be substituted for another in a crushingly advantageous way is what enabled Eratosthenes to do this. The key point being that while we as humans do discriminate between forms, a small circle in the dirt as being something drastically different from a large one that goes around the earth, many of the fundamental rules that we learn about them do not.
A circle is a circle is a circle and once the essential laws have been obtained wherever that is, they apply across the spectrum of forms, from a small circle drawn in the dirt more than 4,000 years ago to a modern circle as large as the very earth itself. It simply makes no difference. p. 70.
Throughout what follows, the often-mentioned computational efficiency of the Hindu-Arabic notation will be on full display. However, there is also much “conceptual manna” to be gleaned from the coin numeral and abacus models that we have developed.
So instead of tossing these systems aside as interesting curiosities and forgetting about them, we will now recalibrate them to become potent weapons of exposition.
These models then will comprise a crucial portion of the conceptual arsenal which we will employ full force in our efforts to give readers a newfound appreciation for the elementary arithmetic already in their possession. p. 72.
Reviews
“The history of numeration is brought to the forefront in How Math Works, A Guide to Grade School Arithmetic for Parents and Teachers. G. Arnell Williams is a huge fan of using diagrams and pictures along with step-by-step breakdowns of each concept he discusses…The examples in the book are simple to understand yet the history is more complex. When reading the book, the reader is sent on a journey through history that is compelling and downright fun. I found myself wanting more.”
— Peter Olszewski, Mathematical Association of America Reviews
“How Math Works delivers exactly what it promises: an extremely thorough explanation of numeration and the four basic operations. It’s completely fascinating, but information-dense and academic...All explanations are accompanied by detailed diagrams and multiple methods: a “coin system” (basically enhanced tally marks), an abacus, and written out in Arabic numerals (though you’ll learn about Roman numerals, too). Learn how ancient Egyptians multiplied using doubling charts. It’s pretty amazing. If you’ve always wanted to understand math but never quite grasped it, this book can help you see how the numbers dance.”
— Tulsa Book Review
“What are numbers? Where did our numbers come from? And why do we calculate with them the way we do? This entertaining, well-written, and instructively illustrated book is a mine of information for anyone (children, parents, teachers, etc.) fascinated by such questions.”
— Robin Wilson, Open University Pembroke College, Oxford
“I don’t know about you but I honestly have to admit that Math confuses me at times! As a Homeschool parent, I find that often it seems like Greek language to me. I have struggled with Math since high school and now that my son is 10, I feel that I need to have a better understanding of Math. Thanks to How Math Works by G. Arnell Williams I do feel like I at least have a better understanding of Math. I love that the book explains the history of arithmetic and the author connects it to other areas such as language which is a subject I love.
“My son enjoyed learning more about numbers and how they came about and I honestly feel that I now am more capable of explaining Math to my son. I’m still not an expert in the subject, but it would take a lot for me to accomplish that goal. I do like that this book was also interesting to read and I highly recommend it to all parents and teachers. Math no longer seems like a complicated and boring subject to me. This book truly has helped me to see the beauty in Math and that is a big accomplishment!”
— Home Education Magazine
“This guide outlines the rich history of math by answering how it formed and how it functions. Williams clearly articulates how math was founded through relationship patterns using symbols such as tally marks, abacus tables, and Hindu - Arabic (numeral) methods. He writes about how necessary these methods are when teaching foundational algorithms in elementary school classrooms. As an elementary school math teacher, I recommend this book for those who are curious about math and are interested in knowing more about its history and how it works.”
— Teaching Children Mathematics
Samples from Book
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 author
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.
pp. xi - xii, How Math Works: A Guide to Grade School Arithmetic…
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.
pp. xiii - xiv, How Math Works: A Guide to Grade School Arithmetic…
Chapter 6: The Symmetry of Repetition
The whole object of travel is not to set foot on foreign land; it is at last to set foot on one’s own country as a foreign land.
—Gilbert Keith Chesterton, British author, literary and social critic
WE EAT THREE MEALS A DAY, go to work five days out of seven, and grocery shop perhaps once a week—repetitive acts are a part of life. Have you ever wondered how many times you do each of these in a year? How would you go about counting something like that? Trivial you say? Think again, for once more, we are close to touching upon timeless principles: touching upon those “special cases” of David Hilbert’s, if you will—the ones containing all the germs of generality.
In this chapter, we will find that figuring out how to conveniently count repetitive actions or situations leads us onto the trail of an entirely new way of reckoning—called multiplication. We begin by considering the following:
A. A bookstore receives a regular shipment of 15 boxes of books every week. How many boxes are delivered in one year?
B. The setup for a graduation ceremony has 30 rows each with 45 chairs. How many guests can be seated at the ceremony?
C. What is the total amount paid back on a signature loan of 36 months where each payment is $115?
D. A 2001 Chevrolet Impala gets 25 miles per gallon. If it has a gas tank which holds 17 gallons, how many miles can it travel when full?
E. A country has a population of 26,784,000 (approximately the same number of people as Nepal in 2011). If the average per capita income is $17,200, what is the combined income in one year for all of its citizens?
These five problems have something in common with our questions about weekly routines; and most people would answer that each can be solved by multiplication. But what exactly does that mean? If a child asked you what multiplication is about, could you explain it? If we knew nothing of multiplication and could only add and subtract, would we still be able to find the answers to these questions?
While each of these problems deals with a different issue, they can be solved by addition alone—provided we add together the number involved in the repetitive act the required number of times. For question:
A. The repetitive act is a shipment of 15 boxes every week for a year (52 weeks in a year); for the answer we must add fifty-two 15s together.
B. The repetitive act is 45 chairs in each of the 30 rows of chairs in the auditorium; for the answer we must add thirty 45s together.
C. The repetitive act is a payment of $115 for each of the 36 months of a loan; for the answer we must add thirty-six 115s together.
D. The repetitive act is 25 miles for each of the 17 gallons of gas in the fuel tank; for the answer we must add seventeen 25s together.
E. The repetitive act is $17,200 for each of the 26,784,000 citizens in the country; for the answer we must add twenty-six million, seven hundred eighty four thousand 17200s together.
The need to repeatedly add or count in multiples occurs in numerous guises.
There are millions upon millions of other situations (an infinite nation of them if you will) that can be linked by the need to repeatedly add numbers together and this fact alone makes this process highly relevant. As with Amar, when he began studying circles: Whatever we discover in our study should have far-reaching, even universal, implications.
But first we need to give chase to the common thread running through all of these problems. What is it? Each of the problems involves a “certain number of objects” (boxes, chairs, miles, or dollars) repeatedly used a “certain number of times.” These two numbers are what we must capture in representing this common thread. In other words, we must identify: the “number of times the repetitive act occurs” and the “number of objects involved in each repetitive act.”
For the situation involving bookstore shipments the repetitive act occurs 52 times and 15 objects are involved in each repetitive act. Commonly used abbreviations linking these two numbers include: 52 × 15 or 52 • 15 or (52)(15). Each of these reads as: add fifty-two 15s together or 52 times 15. In the same spirit, we could rewrite the “total combined income” question as 26,784,000 times 17,200 or 26,784,000 × 17,200. These shorthand abbreviations are written representatives of a new process—one that is different and distinct from simple addition or simple subtraction. We will christen the method with the name “multiplication.”
While we may jump for joy over this christening, with its shorthand notation, it really isn’t very helpful in and of itself. That is, if the only way to calculate 26,784,000 × 17,200 is to literally take twenty-six million, seven hundred and eighty-four thousand 17,200s and add them together, then using this abbreviation hasn’t really given us anything new from the standpoint of saving us work—we still have to do all of the additions to solve problems.
Solving the problem this way is simply out of the question: There are 26,784,000 seconds in 310 days, and it most surely would take a longer time than this to simply count from 1 to 26,784,000—even if that is all one did 24/7. Since it takes more time to add than to count, it stands to reason that performing these many repeated additions of 17,200 in a direct manner (without any shortcuts) would take a great deal longer than 310 days—even using our modern method of doing addition.
Nevertheless this type of problem remains and needs to be solved. What are we to do? We know many of our ancestors got around this problem by using devices such as the abacus to help them multiply. But we have also shown that the HA numerals are a direct capture of a certain type of abacus in writing. Is it possible to now take advantage of this and avoid actually scribbling down a string of numerals the required number of times—making convenient multiplication in writing possible? If so, this would be a bonanza indeed.
In this chapter, we begin an investigation into whether this is possible or not, and will follow G. K. Chesterton’s lead by looking at the familiar territory of multiplication as if we were strangers in a new land.
pp. 99- 102, How Math Works: A Guide to Grade School Arithmetic…
Chapter 7: Dance of the Digits
Dancing in all its forms cannot be excluded from the curriculum of all noble education: dancing with the feet, with ideas, with words, and, need I add that one must also be able to dance with the pen?
—Friedrich Nietzsche, German philosopher, poet, classical philologist
IF THIS BOOK REPRESENTS a journey passing through mountainous terrain, then this chapter represents one of the highest pinnacles that we shall attain. With many peaks, the toughest part of the climb is often the last part. In this vein, this chapter involves a bit more symbolic manipulation than some of the others—but it is still within your purview. I urge the less mathematically inclined of you to stay the course here. We have covered a lot of conceptual terrain to get to this point and now have placed within sight a much deeper understanding of how the modern methods of multiplication taught in the schools really work. Let us not now shirk from the symbols when we are so close. Hopefully you will find the scramble up this last segment of the trail illuminating...
The major focus of the previous chapter was to show that it is possible, in writing, to solve problems that require repeated addition by actually getting around doing all of those additions on paper. This allows us to significantly reduce the time it takes to find solutions in these situations—once more placing before us the universal idea of substituting one thing for another to gain decisive advantages. Due to its widespread occurrence, the name “multiplication” was given to this way of reckoning.
While much progress has been made in taming this operation (the Egyptian method of doubling, for instance), our times table methods in HA script are still not complete. Presently, we can only handle a restricted class of problems—namely, those involving an arbitrarily long number multiplied by a single digit. Computing such multiplications, in principle, has been completely solved and can usually be performed much faster, with times tables, than using the Egyptian technique. But there are many, many other multiplications where both numbers have two or more digits (the five original problems of the last chapter, for instance) and the time has come to learn how to deal with these.
Additionally, we still need to see if the entire process can be miniaturized further into a single diagram in a manner similar in spirit to the compact algorithms we have for addition and subtraction.
Our goals then are this: to develop a general recipe (in HA script) that works for the multiplication of any two whole numbers, not just the special cases of the previous chapter, and then to stylishly simplify it. We will accomplish both in this chapter, and in the demonstration will see the overwhelming power inherent in the symmetry of the HA notation. This symmetry will give us the ability to take the repeated addition of a number and radically transform it into something completely different, into something that in its most expressive form can be likened to a drill team of digits marching to well-scripted rules—a veritable dance of the digits, if you will. This transformation, this poetry of the diagrams, allows for the operation of multiplication, which is significantly harder than either addition or subtraction, to be wrestled out of the hands of the specialist and laid at the doorstep of elementary school-aged children.
Carving Up the Numerals
The plan of attack, when multiplying two numbers in HA script, is to “carve up” the multiplication or product in a way that allows us to unleash the times table on it. When the product is of the form “two or more digits × a single digit,” as in the last chapter, we break up the larger number to accomplish this (e.g., in 62 × 7 we split up the 62 [as 60 + 2] to get 60 × 7 + 2 × 7). However, if both numbers in the product contain two or more digits, we have some freedom in how we choose to decompose. For example, when computing the product 62 × 37, we can choose to carve up the 62 first or the 37 instead. To successfully handle these cases requires that we relocate the trailing zeros in the product.
If a numeral ends in one or more zeros, we will call these zeros “trailing zeros.” For example, 50 has one trailing zero and 500 has two trailing zeros. The numeral 100020 has four total zeros, but only the one at the end qualifies as a trailing zero. In HA script, trailing zeros are like free agents which can roam at will from the tail of one numeral in the product to the tail of the other. The following all show that while the total number of trailing zeros is conserved in the product (before completing the multiplication, that is), they can move freely between either numeral:
One Trailing Zero: 60 × 1 = 6 × 10 = 60
Two Total Trailing Zeros: 60 × 10 = 6 × 100 = 600 × 1 = 600
Four Total Trailing Zeros: 600 × 700 = 60 × 7000 = 6000 × 70 = 6 × 70000 = 420000
Five Total Trailing Zeros: 54000 × 7200 = 54 × 7200000 = 5400 × 72000 = 5400000 × 72 = 388800000
In terms of our viewpoint of multiplication as repeated addition, these seemingly innocuous shifts of zero are actually tremendous simplifications. They have far greater impact on reducing the number of additions required than even does simply reversing the order of the multiplication. For instance, 600 × 700 means to take six hundred 700s and add them together. But by shifting zeros, we can swiftly rewrite this as 6 × 70000 and find the answer by knowing only that 6 × 7 is 42 and then putting the four zeros after it to obtain 420000—meaning in this case, that the collective effect of hundreds of additions has been rerouted to a single calculation that can be completed in less than ten seconds! These shifts play out on abacus rods as shown:
Exploiting the free agent properties of trailing zeros will prove to be pivotal to all that follows in that we can reposition them to convert multiplications between any two numbers, into multiplications found in the multiplication table. We just have to learn how to incorporate all of the shifts in such a way that everything aligns correctly—historically, no small task. If there are still any lingering doubts on the significance of the number zero (and the symbol representing it) to how we perform arithmetic, please let them cease here.
Nontrailing zeros, on the other hand, are bound and cannot be moved around at will since doing so gives different answers. For example, the zeros in the product, 7202 × 6008, are nontrailing and cannot be shuffled around: 7202 × 6008 ≠ 7220 × 6800.
To get the ball rolling let’s look at how the multiplication of “two digits × two digits” plays out by walking through “62 × 37”:
pp. 121- 124, How Math Works: A Guide to Grade School Arithmetic…
