People rarely come to algebra with neutral attitudes. For many, difficulty with the subject in school defines and dominates their mathematical experience, and they often are quite passionate and expressive about it.
Nor are they historically alone in this feeling either. This intensity of emotion is no recent thing.
In 1749, Frederick the Great of Prussia wrote to the famed man of letters, Voltaire, regarding algebra, “… But to tell you the truth I see nothing but a scientific extravagance in all these calculations. That which is neither useful nor agreeable is worthless. As for useful things, they have all been discovered; and as to those which are agreeable, I hope that good taste will not admit algebra among them.”1
More recently, in 1930, nationally syndicated columnist Dr. Arthur Dean wrote, “If there is a heaven for school subjects, algebra will never go there. It is the one subject in the curriculum that has kept children from finishing high school, from developing their special interests and from enjoying much of their home study work.”2
One can find similar sentiments expressed throughout the last four centuries, from the time symbolic algebra was first taught in schools up to this very day.3
So, what then is this thing called algebra? What is it truly about? What are some of the things that distinguish it from arithmetic? And what advantages, if any, does it really offer to those who know how to use it?
These are questions that Algebra the Beautiful aims to shine a discerning spotlight on, tackling them head on from the jump and continuously throughout the book. Going into great creative detail in explaining some of the basic procedures of the subject (such as why do we use letters of the alphabet to describe unknown and variable quantities), it aims to bring back to life the aura and excitement of their discovery and early use while simultaneously explaining in clear, understandable language why they work and some of the capabilities and enhanced perspectives that they can still give to us today.
Algebra has been in its current symbolic form only since the 1600s CE, but algebraic documentation dates back to the Mesopotamian and Egyptian eras (ca. 1700s BCE) and perhaps even into the Indus River Valley civilization—if we could only decipher their writing. Chinese and Greek documentation has also been noted during the first millennium BCE.
Why did it take so long for smart people to make the conceptual leap to the sleek symbolic representations that we see today?
Algebraic historians still are trying to gain a better appreciation of this, but the gap in time clearly suggests that the symbolic representations in use today may not be as intuitive as some think they are, bypassing many important purposes and details. And therein lies much of their amazing efficiency; yet therein simultaneously lies great difficulty, too, for when we teach symbolic algebra to students, it is easy to make a great many assumptions about their understanding of what the symbols and procedures are actually accomplishing. Assumptions that students, who struggle with algebra, are often tripped up by, especially if they are never explicitly pointed out.
One of the central aims of Algebra the Beautiful is to try to get out in front of those hidden purposes and details, making them far more transparent for readers. It does this by focusing on a few accessible topics in great depth and variety. In pursuing this route, the book takes full advantage of the opportunities that open up for exploring nuanced and important matters such as parameters, how the mixing of variation and stability can be gauged through algebraic representation, and the incredible reach and unifying power of algebraic expressions and equations.
The great educational philosopher John Dewey’s thoughts, from his 1934 book Art as Experience, have been described thus:
An experience occurs when a work is finished in a satisfactory way, a problem solved, a game is played through, a conversation is rounded out, and fulfillment and consummation conclude the experience. In an experience, every successive part flows freely. An experience has a unity and episodes fuse into a unity, as in a work of art. The experience may have been something of great or just slight importance.4
Algebra the Beautiful is a math book that seeks to give you many such experiences—experiences that will hopefully transform, for the better, your entire view of the subject of algebra.
It can be likened to tourists experiencing a national park. If the roads are appropriately placed with adequate turnouts, hiking trails, and interpretive centers, visitors can gain a spectacular appreciation of the dramatic scenery, say of the Grand Canyon, without exhaustively visiting every inch of the park.
Algebra the Beautiful aims to make a definitive and lasting impression by showing that algebra uncloaked is big, varied, dramatic, and relevant, forming an interactive, reliable foundation for all of mathematics and many other areas as well. It communicates this to readers through several distinctive approaches, including the following:
• The Humanistic Approach: Algebra (as well as the rest of mathematics) is not an isolated island but rather shares similarities with other great areas of human activity, expression, and ambition—including science, language, history, art, music, and philosophy—that seek to better understand and describe the world and then use this knowledge in impactful ways.5
• The Aesthetic Approach: Mathematicians and scientists frequently state that mathematics is beautiful, yet most nonexperts don’t see it that way. In Algebra the Beautiful, the aesthetic is interwoven into the very fabric of the book. This is done by reimagining elementary algebra as a vehicle for illuminating the general beauty of mathematics. Ideas work together in concert, and paying nuanced attention to the beauty produced by their interaction serves as a powerful weapon of exposition for the book.
• The Conceptual Approach: Metaphors, Narrative, and History: Algebra the Beautiful demystifies the techniques of elementary algebra by using metaphors, analogies, and history in unique and robust ways to tell the subject’s powerful and holistic story. There is magic in the combination.
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.