Book Description
A mathematician reveals the hidden beauty, power, and—yes—fun of algebra.
What comes to mind when you think about algebra? For many of us, it’s memories of dull or frustrating classes in high school.
Award-winning mathematics professor G. Arnell Williams is here to change that.
Algebra the Beautiful is a journey into the heart of fundamental math that proves just how amazing this subject really is.
Drawing on lessons from twenty-five years of teaching mathematics, Williams blends metaphor, history, and storytelling to uncover algebra’s hidden grandeur.
Whether you’re a teacher looking to make math come alive for your students, a parent hoping to get your children engaged, a student trying to come to terms with a sometimes bewildering subject, or just a lover of mathematics, this book has something for you.
With a passion that’s contagious, G. Arnell Williams shows how each of us can grasp the beauty and harmony of algebra.
Passages from Algebra the Beautiful
Algebra the Beautiful doesn’t seek to dazzle you with a stunning display of facts, nor present you with a long list of mathematical formulas.
Rather, the goal of this book is to strike at the heart of your conceptual and emotional understanding of algebra, to put you on more intimate terms with a few of the simple, yet elegant, ideas at the core of the subject while at the same time taking you on an imaginative intellectual journey through mathematics itself.
In short, this book aims to inform, bolster, and inspire your mathematical soul. p. 4.
What algebra does par excellence is sharpen these tools to a much finer point.
Put another way, algebra singularly weaponizes metaphorical and analogical reasoning, rendering it more precise and operational. This is true of mathematics in general, but algebra forms one of the strongest alloys used to forge this mighty mathematical sword. p. 204
A central goal of this book is to learn more about variations of the numerical persuasion and to showcase their accompanying descriptions in symbols. We will find these variations to be relevant, often surprising, and more around us than we might think (often unrecognized).
Moreover, we will find that their systematic description opens wide to us an entirely new and vast-reaching branch of mathematics: one that is distinct and separate from elementary arithmetic on the one hand yet critically fused at the hip with it on the other.
Together these two branches will team up to form one of the most potent one-two punches in the history of human thought—creating, in the process, a quantitative version of Beethoven’s “electrical soil” in which the sibling spirits of mathematics and science can often materialize in, thrive, and discover masterful expression. p. 10.
The types of maneuvers just discussed depend critically on the specific situation at hand. Imagine a scenario in which we could standardize a much larger class of maneuvers, maneuvers that grant us the ability to systematically solve all kinds of seemingly sophisticated and unrelated problems—enabling us to convert some of the elegance and magic of mathematical ingenuity into routine.
In a sense, this is what algebra injects into the mathematical bloodstream: providing a method to reduce the brilliant and extraordinary into the ordinary and reproducible. p. 31.
The realization that many things which don’t look or seem alike at all can still be tied together by common mathematical expressions, equations, and reasoning is a key ingredient in becoming more mathematically aware.
This, combined with the knowledge that in many cases the only way to make such connections at all appears to be through mathematics, demonstrates that there is a lot going on out there in the world that we simply can’t see without tuning into mathematics—“to fly where before we walked,” as Bill Thurston proclaimed. p. 159.
Reviews
“G. Arnell Williams has undertaken a challenging job — to show the importance, deep structure, intellectual connections, and sheer beauty of classroom algebra. He doesn’t shy away from questions like why, or even whether, we should teach algebra in high school…
Elementary algebra has in it two powerful ideas, whose importance to human progress is hard to overestimate. What are these two ideas? “Let x equal” and “For any a and b.”…
The power and beauty of algebra comes from uniting these two ideas —general symbols and the finding of unknowns. And this is what the heart of Williams’s book is about.”
— Judith Grabiner (fellow of the American Mathematical Society), Journal of Humanistic Mathematics
Author G. Arnell Williams demystifies algebra in Algebra the Beautiful. The book shows that algebra is not an isolated subject…The book’s opening chapter begins by demonstrating that sometimes arithmetic is not enough. The language of arithmetic allows us to determine a solution for just one particular problem.
Whereas, the language of algebra allows us to determine and express a multitude of solutions, not just for one single problem but for a whole class of similar problems…
Algebra the Beautiful includes a gentle history of the invention of algebra. The reader is treated to a light review of Diophantine equations, the work of Al-Khwarizmi, Viète, … and host of other contributors to the development of algebra.
The reader is also treated to enlightening historical discussions of the pros and cons of introducing the subject of algebra into both high schools and universities in the United States. These historical accounts are perceptive and thorough.
— Tom French, Mathematical Association of America Reviews
“Algebra the Beautiful is a rich paean within a workbook (of sorts), or possibly a workbook within a paean…. Mr. Williams articulates and affirms his goal, admittedly ‘lofty, illusory’—yet attainable, he believes: teaching algebra through fulfilling experiences and unifying ideas.”
— Siobhan Roberts, Wall Street Journal
“Math professor Williams (How Math Works) successfully makes his case that algebra is “big, varied, dramatic, and relevant” in this shrewd attempt to win over math-averse readers. Using what he terms humanistic, aesthetic, and conceptual approaches, he relates the field to “other great areas of human activity, expression, and ambition.”
“The book is interesting, parts are beautifully written, and the whole is quite original.”
— David Mumford, Fields Medal laureate, and author of Indra’s Pearls
“G. Arnell Williams has written a book that alters the way students who struggle with algebra should learn the subject. Algebra the Beautiful embraces the foundational contents of a few approachable topics that link to a further complete and deeper understanding of algebra.”
— Joseph Mazur, professor emeritus, mathematics, Marlboro College, and author of Euclid in the Rainforest
“Algebra, a dreaded subject for many students, is the focus of this popular-science book by G. Arnell Williams, a math professor. Recognizing that many adults still struggle to understand it, Williams adopts a narrative approach aimed at nonspecialists to demystify algebra’s concepts and rules.”
Samples from Book
Chapter 1: Numerical Symphonies
Music is the electrical soil in which the mind thrives, thinks and invents.
— Ludwig van Beethoven (1770–1827), letter from Bettine von Arnim to Johann Wolfgang von Goethe
Algebra is a vast and beautiful continent—at times serene and familiar, at other times mysterious and wild.
Despite the fact that it has been powerfully used for centuries, underwriting some of humanity’s most important innovations, serious questions and riddles remain: especially in regard to its essential nature, its place in education, and why so many intelligent people struggle to understand it.
Consider this an invitation to experience some of the vastness, aura, and beauty of this terrain.
But algebra does not reveal its scenery for free. It requires an intense cocktail of conceptual techniques to bring this beauty into sharp relief.
Consequently, I will heavily employ some of the most powerful weapons of exposition available including metaphor, analogy, history, and narrative.
Of metaphor, mathematics education researcher Anna Sfard states:
Metaphors are the most primitive, most elusive, and yet amazingly informative objects of analysis. Their special power stems from the fact that they often cross the borders between the spontaneous and the scientific, between the intuitive and the formal. Conveyed through language from one domain to another, they enable conceptual osmosis between every day and scientific discourses, letting our primary intuition shape scientific ideas and the formal conceptions feed back into the intuition.
Of history, mathematician J. W. L. Glaisher said, “I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history.”
And of narrative, cognitive psychologist Steven Pinker says, “Cognitive psychology has shown that the mind best understands facts when they are woven into a conceptual fabric, such as a narrative, mental map, or intuitive theory. Disconnected facts in the mind are like unlinked pages on the Web: They might as well not exist.”
Many experts share these sentiments.
In this book, we explore how far we can go with injecting these techniques (with a vengeance) throughout the discussion. My hope is that it will transform your conceptual and emotional understanding of this oft-maligned subject. In this chapter, we begin with music.
MUSIC
Music is one of the most remarkable of all the activities of humankind. Millions willingly subject themselves to its mood-altering effects day after day. Just a simple thirty-second ditty or jingle can launch back to life memories from decades past.
Mute the sound to a video of people vigorously dancing and their energized behavior looks fascinating at best, bizarre at worst. Go to the mall, ride an elevator, or watch a movie, and you will find it there lurking in the background. It is everywhere.
But what exactly is it? Why does it impact people in the ways that it does? How can it launch some into a state of almost pure euphoria while reawakening painful emotions, long thought extinct, in others?
It too is a vast expanse of familiarity, serenity, and mystery.
Of all its forms and manifestations, one of the most grand, vivid, and complex arises in the guise of the symphony: “a lengthy form of musical composition for orchestra, normally consisting of several large sections or movements . . .”
The trajectory of sounds in a symphony can be extensive, wide-ranging, and dramatic. Reaching a profound and notably intense form in the work of Ludwig van Beethoven, it is said to be the medium in which many composers still choose to demonstrate their technical prowess and most expressive ambitions.
Our interest with it here lies primarily in the great variety and scale that can arise around a central well-developed theme (or core)— we will see something similar happen repeatedly in a mathematical context.
One of the most fascinating things about music is that it is possible to capture the dynamic range of a symphony on flat sheets of paper. It is almost as if musicians can freeze the essence of an hour’s worth of lively music and hold it in suspended animation to be viewed later and analyzed at their leisure. This is no small thing and offers great benefits to those who choose to use it.
Written musical notation gave Beethoven (who could barely hear at all by his forties) the inspirational capacity to compose and share wonderful music right on up to the last years of his life, with the release of his highly acclaimed Ninth Symphony occurring in 1824 at age 53. It is hard to imagine him doing this without the aid of visual notation. To this very day, orchestras are still able to perform these masterpieces thanks, in large part, to their preservation in written form.
But the sounds of music in a complex performance are not the only phenomena that can vary in our world.
pp. 7 - 9, Algebra The Beautiful
Chapter 2: Art of Maneuver
The mathematics of our day appears to me like a large weapon shop in peace time. The store window is filled with showpieces whose ingenious, artful, and pleasing design enchants the connoisseur. The real origin and purpose of these things, to attack and defeat the enemy, has retreated so far into the back-ground of consciousness as to be forgotten.
— Felix Klein (1849–1925), Development of Mathematics in the 19th Century
In walking, it means going around the muddy puddle to get to the store dry and clean. In photography, it entails looking for the best vantage point to make a sunset sing. To sports goers, it can refer to leaving the game early to avoid traffic. To running back Barry Sanders, it meant getting around defenders in ways that were almost choreographic.
Maneuver is all around us in a wide variety of manifestations. In some domains, its presence is so pervasive and overwhelming that its very name simply cannot be hidden from view—taking on the mantle of entire doctrines even. Military thinking is one such arena, where the original meaning of the word was closely allied with the notion of moving forces on the ground into favorable positions that hastened the defeat of the enemy.
In its landmark philosophical document, Warfighting (1989), the US Marine Corps gives the following description: “maneuver warfare is a philosophy for generating the greatest decisive effect against the enemy at the least possible cost to ourselves.” Methods to achieve this effect now also include deception, surprise, shock, and speed not only on the ground but from the air and from the sea.
Two of the foremost military thinkers in history, Sun Tzu and Carl von Clausewitz, spend attention to the idea in their respective famous works, The Art of War and On War. It remains a hotly debated and energetic topic in strategic circles.
The word maneuver, however, is quite versatile in its other wide-ranging uses and definitions, among them “an action taken to gain a tactical end,” “a clever or skillful action or movement,” and “doing something in an effort to get an advantage or get out of a difficult situation.” Our use of the word in this book will contain aspects of each of these definitions. For our purposes, we summarize with the following general description:
Symbolic maneuver includes any introduction, combination, movement, and/or manipulation of symbols (including diagrams) to gain an advantage in knowledge, insight, organization, clarity, efficiency, etc.
Undoubtedly both too broad a description and simultaneously not comprehensive enough, this definition will serve the good-enough purpose here to characterize what is one of the central and foundational features in algebra. In this chapter, we place a magnifying glass on this cornerstone idea.
MANEUVER IN ARITHMETIC
Symbolic stratagems to gain an edge extend beyond mathematics. The use of symbols in language can also be looked upon as a form of sophisticated maneuver. When we tell someone about a vacation last summer photographing waterfalls in Wells Gray Provincial Park (British Columbia), we are actually using language symbols to get around limitations.
We cannot physically re-create the waterfalls, forests, rivers, mountains, wildlife, adventures, and interactions with people that we experienced on the trip, but we can symbolically share them with others using the words we give them in language—to tell stories of our experiences.
Similarly, many of the techniques and algorithms we employ in elementary arithmetic can be looked upon as symbolic maneuvers.
Consider multiplication: How much in ticket sales might we expect to earn from an event that is scheduled to be attended by 965 people at a cost of $175 per person? We can obtain a reasonable estimate of what the revenue should be by simply finding the answer to 965 × 175. One way to get this would be to take nine hundred sixty-five 175s (one for each attendee) and add them together:
Though we can obtain the answer this way, almost no one would do so; it is simply too slow and far too painful. What we generally do is take the information (965 × 175) and maneuver it into another form. A millennium ago, folks might have used some sort of device such as the abacus or counting board to find the answer, but today we have several options.
We could take the numerals and directly key them into a calculator and have the machine do the multiplication for us, using procedures coded in electricity. Or we could reformat the information in a way that allows us to perform swift moves in writing (with the aid of a multiplication table) like so:
This is a symbolic maneuver that keeps us from having to do anything close to the original 965 additions—conveniently showing us that the revenue should be $168,875.
pp. 27 - 29, Algebra The Beautiful
Chapter 4: Converging Streams and Emerging Themes
The essence of knowledge is generalization. That fire can be produced by rubbing wood in a certain way is a knowledge derived by generalization from individual experiences; the statement means that rubbing wood in this way will always produce fire. The art of discovery is therefore the art of correct generalization. . . . The separation of relevant from irrelevant factors is the beginning of knowledge.
— Hans Reichenbach (1891– 1953), The Rise of Scientific Philosophy
In the background of any area of mathematics lurks a most remarkable and pervasive presence. Mathematician Morris Kline hints at it when he writes:
. . . A study of mathematics and its contributions to the sciences exposes a deep question. Mathematics is man-made. The concepts, the broad ideas, the logical standards and methods of reasoning . . . were fashioned by human beings. Yet with this product of his fallible mind man has surveyed spaces too vast for his imagination to encompass; he has predicted and shown how to control radio waves which none of our senses can perceive; and he has discovered particles too small to be seen with the most powerful microscope. Cold symbols and formulas completely at the disposition of man have enabled him to secure a portentous grip on the universe. Some explication of this marvelous power is called for.
Whatever the full extent and nature of this “marvelous power” of mathematics, traces of its presence should be visible even in the most fundamental areas of the subject. As these are the areas where we reside in this book, part of our task in exploring algebra should be to give a glimpse of this potency in its pages. A few observations are the order of the day.
Imagine you knew nothing about mathematics and were playing around with pebbles on the ground. It would still be possible to learn basic facts, such as two pebbles added to four pebbles yields six pebbles, or taking seven groups of three pebbles together yields a total of twenty-one pebbles. In fact, it would be possible to learn a great deal about arithmetic from tinkering around with pebbles only.
But the wonderful thing then is that what is true of pebbles is true of so much more, for if you replace the pebbles with people, dollars, or cars instead, you still obtain true results. That is, it is not just true that 2 pebbles + 4 pebbles = 6 pebbles or that 7 × 3 pebbles = 21 pebbles, but it is also true that 2 people + 4 people = 6 people and that 7 × 3 people =21 people—and the same with dollars, cars, and so many more things.
In short, by playing with pebbles, one can not only discover arithmetic facts that are true of pebbles, but one can in effect discover arithmetic facts that are true of infinitely many things! It is almost as if the universal laws of addition and multiplication in arithmetic momentarily materialize in the guise of the pebbles in your possession, uncloaking some of their general properties to any and all who would take notice.
Moreover, this remarkable ability to generalize from a specific experience is not unique to mathematics. We may learn the rules for how to safely cross a street on one near where we grew up, yet those same basic principles and behaviors can be used to successfully cross a similar street on the other side of town, in Waterbury (Connecticut), in Trondheim (Norway), or on practically any similar street on the face of the Earth!
From learning how to play an instrument, a board game, a video game, or a sport to learning how to type, drive a car, or cook, this ability to project our knowledge and skills from the context in which we first acquired them, and extend them to millions of other similar circumstances, is everywhere. Amazingly, we all have been granted the ability to connect our individual experiences to something much greater than ourselves—touching, in a sense, eternity itself from the confines of home.
Perhaps the most compelling manifestations of this phenomenon occur in the traditional scientists’ and inventors’ laboratories, where breakthroughs such as the telephone, the light bulb, X‑rays, radioactivity, and the transistor first were discovered or developed in the small, then have ultimately come to affect millions and in some cases billions of people in the large. And this ability to flirt with the eternal from home is also present right here in elementary algebra, where the time has come to spend more of the algebraic capital that we acquired in the first three chapters. Here, we begin with a simple situation from business in the small. Let’s see where it takes us.
MONEY STREAMS
Let’s consider a situation involving a newly formed small business venture where hamburger meals will be sold. After doing a bit of research, we have determined that overhead (rent, maintenance, utilities, etc.) every two months will cost around $3100. We have also learned that it will cost about $4.75 for the time and ingredients to make each hamburger meal. Based on this information, we have settled on a selling price of $6.00 for each meal. This guarantees that we will make more money from each sale than we spend.
What we would like to know is how many of these meals we have to sell every two months so that the amount of money we bring in (revenue) matches the amount of money we spend (total costs). For the purposes of simplifying the problem, we will make the assumptions that our cost estimates are exact and that we sell all of the food we buy.
To get a feel for the problem, we will start by playing with a few hypothetical situations. What happens, for instance, if we were to make and sell 100 meals? We have two cases to consider: the amount of money brought in and the total amount spent. Based on the conditions just listed, these are both straightforward to obtain:
So, we see in this situation that the revenue is $2975 less than the total costs ($3575 – $600), which is how far we still are in the hole. But this is already an improvement over the $3100 overhead debt we started with before making and selling any meals.
What if we make and sell 400 meals?
Here, we see that both revenue and costs have grown, but the gap is closing as the difference between them now is only $2600 versus $2975 from the first situation.
We continue our testing for the situation where we make and sell 1000 meals:
pp. 83 - 86, Algebra The Beautiful
Chapter 6: The Grand Play
By whatever means it is accomplished, the prime business of a play is to arouse the passions of its audience so that by the route of passion may be opened up new relationships between a man and men, and between men and Man. Drama is akin to the other inventions of man in that it ought to help us to know more, and not merely to spend our feelings.
— Arthur Miller (1915–2005), Introduction to Collected Plays
To acquire an intellectual advantage at great cost, if it can be attained more cheaply is unnatural . . .
— Carl von Clausewitz (1780–1831), Clausewitz and the State
On Monday, January 6, 1930, the great American philosopher and education theorist John Dewey was invited to give a series of public lectures on philosophy and psychology at Harvard University. He accepted on January 13, and one year later, he gave a sequence of ten talks so extraordinary that their content was later collected in a book called Art as Experience (1934).
Scattered among the book’s sometimes dense passages are conceptual gems of such dimension, creativity, and zing that they continue to rouse the imagination:
Every art communicates because it expresses. . . . For communication is not announcing things. . . . Communication is the process of creating participation, of making common what had been isolated and singular.
A primary task is thus imposed upon one who undertakes to write upon the philosophy of the fine arts. This task is to restore continuity between the refined and intensified forms of experience that are works of art and the everyday events, doings, and sufferings that are universally recognized to constitute experience.
We have an experience when the material experienced runs its course to fulfillment. . . . A piece of work is finished in a way that is satisfactory . . . is so rounded out that its close is a consummation and not a cessation. . . . The experience itself has a satisfying emotional quality because it possesses internal integration and fulfillment reached through ordered and organized movement.
Dewey advances the view that many aspects of everyday life have the potential to be aesthetic experiences, and that they should be recognized as having this capacity. He felt that artistic activity is not exclusively the domain of the fine arts, where it is intentionally created to be experienced—as enlightenment captured or tamed—often in museums or galleries, but rather that the aesthetic experience can also happen as events unfold and unite in wild, free-flowing forms.
The artistic experience comprises a continuum between its fixed forms on the one hand and the high points of everyday life on the other, where actions and ideas converge together in unison: the smooth execution of a well-crafted, victorious game plan or campaign strategy, the successful, mature handling of a stressful task, the beautiful rendition of a song at graduation, the enjoyment of a long-desired vacation, or even a crossword puzzle or sudoku grid completed in a creative way after some struggle. Such peak experiences also include the epiphanies that occur in education: namely, moments of insight where new concepts and old memories converge together in students so that they suddenly see or understand something in a strikingly fresh or very clear way.
Dewey thought that instructors should leverage such moments in their teaching and deliberately engineer scenarios to assist students in experiencing the excitement and satisfaction of substantive understanding along with the joy of insight, which has been described as being “a sense of involvement and awe, the elated state of mind that you achieve when you have grasped some essential point; it is akin to what you feel on top of a mountain after a hard climb or when you hear a great work of music.”
To date Dewey is still recognized by many as the preeminent education thinker of American origin.
CONCEPTUAL FUELS
The great acting teacher Stella Adler shared some of Dewey’s sentiments about unity and continuity in her own field, telling her students, “Your curse is that you have chosen a form that requires endless study. . . . It means you have to read, you have to observe, you have to think, so that when you turn your imagination on, it has the fuel to do its job.”
Widely considered to be one of the twentieth century’s leading teachers of drama, Adler believed that imagination and research rather than personal memories or emotional recall should inform an actor’s craft, and that much of what an actor learns, observes, and thinks about can serve as fuel for their performances. It is a powerful statement because she is not suggesting passivity in having this happen either, but rather appears to advocate that actors actively and aggressively employ their individual point of view—their observations, thoughts, and imagination—as conceptual fuel for acting.
Quotations serve as fuels for the imagination, too. They are very popular, and many people are known to inventory collections of them for study and use as well. Why do they do this? Some do it for fun, but I dare say that a large portion also do it for the down-the-road purpose of later using the quotations to motivate, inspire, or enlighten others (or their future selves) for purposes or situations that may have been unknown to them at the time that they first decided to save the quote.
The thing that makes quotes so appealing to us is due in part to the fact that situations in life rhyme and sometimes an apt demonstration of a relationship or a phenomenon in another area (or at a different time) may be encapsulated in a quotation. The statement may provide a spot‑on conceptual perspective or game-changing orientation to a more abstract, unpredictable, or confusing situation. Quotations communicate, across the ages, other people’s singular moments and insights—serendipitous times where they have been gifted with the ability and vision to snatch something transcendent and eternal from a profound, imaginative, or confluent instant in time or thought.
When such a statement powerfully resonates with a person, it is almost like a window in time opens up for them to a wider world of shared experiences, allowing them to momentarily glimpse, “in the palm of their hand,” some portion of the past, present, and future all at once. These junctures become like individual little packets of Dewey-type experiences in and of themselves. And their preservation is a spectacular thing to have available at our fingertips. People’s use of quotations as fuel for their own interpersonal interactions and private inspirations approximates what Adler is telling her acting students to do in employing the results of their “endless study” to fuel their various performances. It also approximates the conceptual fuel that algebra can supply in providing greater insight into certain types of quantitative situations.
WHAT TO DO ABOUT WORD PROBLEMS
Dewey’s and Adler’s thoughts are highly relevant to the case for algebra in general, and algebraic word problems in particular…
pp. 125 - 128, Algebra The Beautiful