__MOVEMENT 6__

To be emotionally stirred is to care, to be concerned. It is to be *in* a scene or subject not outside of it… The more anything, whether an object, scene, idea, fact or study, cuts into and across our experience the more it stirs and arouses…

No amount of possession of facts and ideas, no matter how accurate the facts in themselves and no matter what the sweep of the ideas—no one of these in themselves secure culture. They have to take effect in modifying emotional susceptibility and response before they constitute cultivation.

—John Dewey (1859–1952), Address to Harvard Teacher’s Association, March 21, 1931

__12__

We must look to our own faculty for discerning those fine connective things—community of aim, interformal analogies, structural similitudes—that bind all the great forms of human activity and aspiration—natural science, theology, philosophy, jurisprudence, religion, art and mathematics—into one grand enterprise of the human spirit.

—Cassius Jackson Keyser (1862–1947), “The Humanization of the Teaching of Mathematics,” *Science*

Our brains are complicated devices, with many specialized modules working behind the scenes to give us an integrated understanding of the world. Mathematical concepts are abstract, so it ends up that there are many different ways they can sit in our brains. A given mathematical concept might be primarily a symbolic equation, a picture, a rhythmic pattern, a short movie—or best of all, an integrated combination of several different representations…

A whole-mind approach to mathematical thinking is vastly more effective than the common approach that manipulates only symbols.

—William P. Thurston (1946–2012), Foreword to *Crocheting Adventures with Hyperbolic Planes* and Foreword to *The Best Writing on Mathematics 2010*

“Every thing throws light upon every thing,” observed the “Yale Report of 1828”—the nineteenth-century’s strongest bulwark for retaining what has been called the classical curriculum prescribed for all students attending college.__ ^{1}__ As discussed in

1828 Yale University Catalog Course of Instruction (pp. 24–25): “Livy, three books” is Roman history, “Adam’s Roman Antiquities” is Roman manners and customs, “Graeca Majora” is an anthology of canonical scholarship in Greek, and “Fluxions” is calculus

The “Yale Report” was the Yale faculty’s passionate attempt to justify continuing with this curriculum, which in Yale’s case wasn’t 100% totally wedded to the far-distant past as was sometimes claimed—with the faculty having added, for example, symbolic algebra, logarithms, calculus, and political economy to its roster, which were relatively modern fields of study for the era. But a large portion of it was based on antiquity, and it certainly was a rigorous course of study to most students of the day, with many, if not a majority, growing to eventually despise it.__ ^{3}__ The more recent mathematical courses in the curriculum only compounded the difficulty, and people were quite justified in questioning the relevance, arrangement, difficulty, and manner in which the entire course of study was taught: just as they often are with the curricula of today.

The Yale faculty contended that the essence of the classical curriculum spoke to many recurring features of professional life, and though not always directly applicable to other areas, the broad principles, techniques, and “strenuous exercise of the intellectual powers” honed by this course of study embraced what was “common to them all.”__ ^{4}__ Although the classical curriculum of 1828 eventually gave dramatic way to a far more diversified set of course offerings and majors by the century’s end, its most idealistic leanings of using a broad approach to education—through illuminating and exploiting the connections between a wide array of disciplines and life in general—is still alive and well in some circles today.

And here is the same desire, embedded in the assertions cited at the beginning of the chapter by Keyser and Thurston, though they are separated by nearly a century. The mathematician Alvin White called the application of this philosophy to mathematics “humanistic mathematics” and helped to jump start a robust network for this way of thinking about math as a human endeavor in the late twentieth century and beyond. The refrain behind this way of thinking is that many important rhythms of life as experienced both individually and collectively by human beings manifest themselves in a wide array of subjects—including mathematics—and that these different manifestations should be deliberately sought out, identified, and connected together. This pedagogical philosophy represents one of the grandest of all the grand confluences in education, acting much like the conceptual superhighway from __Chapter 8__ to connect big ideas across a wide range of disciplines—both inside and outside of the curriculum.

Though proponents and critics alike believe that an educational curriculum at any level should have a significant impact on the students passing through it, they vehemently disagree on the particulars of what that curriculum should be or how to devise it. Specifically in the case of algebra, some passionately say that the significant and impactful thing that is happening to students nowadays is actually in reverse of what it should be, believing that it harms those who are forced to study a subject that will be largely irrelevant to their professional lives and may prevent them from advancing or completing their education. They remain firm in the stance that it simply must be cast aside as a general requirement, and replaced by something else.

Others defend algebra’s current place in secondary and adult education and the methods used to teach it, insisting that it is a necessary standard to develop students’ technical capacities, and that the problem with algebra today is a dearth of qualified teachers, not the subject itself or standard instructional methods.

Still a third group agrees with the second that algebra should remain a required subject but is sympathetic to the concerns of the first, arguing for reforms to address the serious philosophical and infrastructural problems in math education, such as a mountain of topics and an overemphasis on manipulating symbols, testing, and irrelevant practice problems. In their view, simply finding better-qualified teachers is not enough to fix a fundamentally flawed situation, and we should focus our energies on developing new and compelling pedagogical methodologies that make the subject far more understandable and appealing to the majority of students required to take it—methodologies that reflect the true potency and beauty in the subject. This is also a hope often expressed by more than a few mathematics professionals over the past two centuries.

To use a crude military analogy, the first group’s stance is somewhat akin to abandoning troop positions in ambiguously held territory and redeploying them to more secure locations, in the belief that adequate supply lines do not exist and can never be established. Thus, making the current positions too dangerous for the majority of troops, save, perhaps, for a comparatively few specialized units. Those who argue for improving status quo methods can be viewed as wanting to maintain these positions in the belief that adequate supply lines do exist, but are failing due to incompetent or inadequately trained staff and troops. The third camp, advocating for reform, accordingly believe that by reinforcing the troops and creating additional, alternate routes within a preexisting broad network of supply lines, their position need not be abandoned—and, moreover, can be strengthened. Keyser’s and Thurston’s statements most naturally align with those in the third camp.

Viewed purely from a military point of view, all three possibilities may make sense depending on the circumstances in which an army in the field finds itself. But what of the educational circumstances that legions of students find themselves facing in an algebra class?

I have adopted the viewpoint of the third camp and have devoted a substantial portion of this book to opening such pathways into existing supply lines for algebraic illumination, insight, and understanding. In this concluding chapter, we’ll summarize some of the forms that these paths have taken, but it remains for you to judge for yourself whether these attempts have been successful.

**RELATIONSHIPS, SYMBOLS, AND MANEUVERS**

Just as arithmetic gives us a precise quantitative vocabulary to express notions such as one collection being larger than another collection, algebra allows us to give shape and personality to our interactions with numerically variable quantities. This is not insignificant, taking full advantage of certain capital features of nature. Consider that though one can write the name of a town with pencil, ink, or chalk, it is also possible to spell the same word using toothpicks, bricks, shrubbery, contrails from an airplane, sand, tape, dominos, people, Play-Doh, and so on. Although its letters can be formed from dramatically different materials—even those not designed for writing—the order of the letters and relationships between them are paramount and are what continue to allow us to recognize the word.

Relationships that we establish between objects, concepts, and quantities can translate to a broad spectrum of materials and circumstances, enabling us to connect them to a broader network of ideas and scenarios. Algebra is inherently relational, helping us to express, maneuver, and transform a wide array of quantitative relationships in extremely efficient and often emergent ways. Two of the most important ways in which algebra does this have been characterized in this book as the two dramas, with the first drama roughly corresponding to what can be captured by variable expressions and the second drama corresponding to what is capturable as equations. A third way—involving graphical representations—has not been critically examined in this book, but is nevertheless extremely important in its own right.

In the first two dramas, relationships between numerically varying quantities are translated into symbolic relationships between the letters that represent them, thus rendering variable situations into a visual and operational format. This represents another grand confluence because once we take variable phenomena and capture them through the lens of visible symbols, it allows us to sometimes make surprising connections in algebraic writing between situations that on the surface may not look similar at all, as occurred in the section “Algebraic Songs” in __Chapter 7__ and throughout __Chapter 8__.

These translations to symbolic representations can be likened to the way spectral analysis allows us to identify the constituent material components of astronomical objects by analyzing the light they emit or absorb—or to how detectives can link a person to the scene of a crime by comparing the unique ridges and curves of their fingerprints with those at the scene. In algebra, it is the visible symbols and the relationships they express that facilitate these connections. The transfer of variable situations into their notational renderings really do serve as a type of algebraic fingerprinting.

But algebra offers more than just the symbolic fingerprinting of variable behavior. Consider that, of all the physical materials we mentioned to form a word, the Play-Doh variant is probably the most pliable and easiest to manipulate. Play-Doh has memory and can represent and hold to a certain shape, but it is also malleable and can easily be molded into other shapes. Algebra and arithmetic as we practice them today are similar insofar as the symbols we use can both hold a certain form and also be conveniently reconfigured into other useful forms.

For instance, in arithmetic, we may have six collections of 22, 57, 28, 45, 23, and 25 items, respectively, whose sizes we are able to write down with numerals—and that may be all that is needed, depending on what we want to do with the information. In this case, the symbols act as an aid in remembering the amount in each collection. However, these numerals contain far more content than just the ability to record information. As we know, they can be combined and maneuvered to give new insights. We can start by setting up the addition of 22 + 57 + 28 + 45 + 23 + 25. This can, of course, be calculated in the traditional vertical fashion, but we can also reorganize the problem as (22 + 28) + (57 +23) + (45 + 25), which allows us to even more quickly condense the expression into 50 + 80 + 70 to obtain 200. This computational feature, where the numerals transform—according to certain well-established rules—into new values that match what we actually observe, is an example of when the symbols act in a malleable fashion.

It was a revolutionary discovery in arithmetic that certain types of numerals could actually be made to conveniently and reproducibly transform in this way. That is, these numerals were not static vessels for storing information, but dynamic vehicles that could be maneuvered by users into making nontrivial, extensive calculations singly on their own in writing unassisted by a mechanical instrument. For most ancients, notations such as Roman numerals and hieroglyphic numerals were primarily used to record quantitative information with the calculations being performed on a computing device like an abacus, but the ancient Indians found a way to unify both information storage and calculational functionality into one set of numerals—the Hindu-Arabic numerals—and the rest, as they say, is history.

A similar and equally dramatic revolution occurred in algebra, but it took mathematicians centuries to come to this realization about the possibilities with the symbols used there, with the old rhetorical methods expressed in words giving dramatic way to the more operational symbolic ones developed in the watershed sixteenth and seventeenth centuries.

Firstly, modern symbolic algebra inherits all of what representational and computational arithmetic offers, then scales and intensifies it to represent infinitely many possible versions of an arithmetic statement or action by a single grand variable expression. Secondly, it allows us to capture and separate out all of the different types of variations in a situation, combining those that are the same and separating those that aren’t, simultaneously managing variation on multiple channels. When handled properly, these procedures of algebra become strategic and decisive deployments not just pedestrian manipulations, all of which imbue algebra with the power to process and organize the symbolic representation of many types of variations that occur in nature—rendering it indispensable to fields such as science, business, statistics, and data science.

**AT THE ALGEBRAIC SPORTS BAR**

On Saturday afternoons in the fall, sports bars across the nation may have on as many as a dozen or more different Division I college football games, each one flavored by the history, traditions, and unique fan bases of both teams on the field. Some of these games will be remarkably one-sided affairs, as a powerhouse team takes on a much weaker opponent, whereas others will go down to the wire and may even result in an upset.

Yet, despite the variety of possible matchups and outcomes, all of the games will still have enough in common that we can easily recognize them all for what they are—college football games. Changing channels from game to game, we can see that in spite of the variation between contests, they each feature 60 minutes of regulation play, a system of downs, a line of scrimmage for each play, players clad in protective helmets and pads with numbered jerseys, referees, a brown ball with pointed ends, and other distinctive rules and positions. The simple ability to change TV stations from game to game, or to view them on multiple screens, provides a sports example of the dynamic interplay that is possible between variable elements at work around a strong core of stability.

This is but one illustration of many such productive interactions between variation and stability that occur in many aspects of nature and society—others include the following:

• **Spoken languages:**

° **Variables:** There are more than 6000 spoken languages in active use worldwide.^{5}

° **Constants:** Languages share the common goal of facilitating structured communication with words that convey meaning in speech, gestures, or writing.

• **People:**

° **Variables:** The global human population numbers in the billions, but each person is an individual with a unique personality, physical characteristics, talents, skills, and ambitions.

° **Constants:** Humans are characterized by common anatomical features, including a complex brain, heart, bipedalism, and opposable thumbs, among other mammalian traits.

• **Nations:**

° **Variables:** There are nearly 200 different nations in the world.

° Constants: Every nation has a territory, population, laws, and a system of government.

• **Automobiles:**

° **Variables:** There are hundreds of different models of automobiles in service today.

° **Constants:** Automobiles share many common characteristics, including an engine or motor, energy source, brakes, and tires.

Sometimes, this intricate dance between variability and stability can be of a much more quantitative nature, which can help us articulate more detailed and precise mathematical descriptions. In this book, we’ve discussed examples that include

• businesses, with varying costs and revenues, yet common methods to represent and locate their break-even situations;

• course instructors, with varying assignment categories and weights, yet common methods to represent and compute their course grade averages;

• students, with varying course loads and academic performance, yet common methods for schools to compute their grade point averages;

• investors, with portfolios of diversified investments at varying levels of appreciation or depreciation, yet common ways to calculate their overall return on investment.

Algebra provides a powerful way to comprehensively treat these latter quantitative situations, using symbols to organize and coordinate the variable components and the more stable ones. We’ve identified these more consistent or stable components in this book as scenario variables—stable within a given scenario, but variable from scenario to scenario.

The existence of scenario variables and the need to distinguish them from traditional unknowns was probably the most valuable of the major advancements in algebra during the 1500s—which is saying something, as it was a banner century for algebra. As we discussed in __Chapter 4__, scenario variables are more commonly called parameters and were definitively introduced in the late sixteenth century by French mathematician François Viète.

A little over 40 years later in 1637, French mathematician and philosopher René Descartes modified Viète’s ideas in his treatise *La Géométrie* by using letters early in the alphabet for parameters and letters later in the alphabet for traditional unknowns, or regular variables.__ ^{6}__ Though Descartes protocol is often followed in elementary algebra instruction, it can be much more inconvenient to adhere to in ensuing applications, as we saw in

These two types of variable quantities—parameters and regular variables—allow us to transform single-instance algebra into what we have termed “big algebra,” illustrating how algebraic insight gleaned from one problem can be scaled to a wider range of applications. For instance, to find the break-even point for various businesses, we may have to solve an individual equation such as 5*x* = 3*x* + 20000 or 752*x* = 534*x* + 987000 or 15.65*x* = 7.82*x* + 72000. As we witnessed firsthand in __Chapter 4__, parameters then enable us to represent all three of these equations, and thousands more like them, by the single all-encompassing equation *Px* = *Cx* + *F*, where *P*, *C*, and *F* take on particular constant values in the context of a specific problem, but change values to reflect different scenarios. This single equation with parameters establishes a holistic, structural connection between all break-even problems that share the same constraints, and individual equations specific to a given scenario become particular instances of that general equation. For example, in the first and second cases, respectively, we have *P* = 5, *C* = 3, *F* = 20000 and *P* = 752, *C* = 534, *F* = 987000.

We considered a similar case in __Chapter 8__ where we saw that for the three assessment categories of student homework, tests, and final exam scores, we could represent all course average calculations with the single big algebra formula *ax* + *by* + *cz*. Here, the letters *a*, *b*, and *c* serve as parameters and capture the contribution values—or weight—of each category, which remain consistent for a particular instructor in a given class, but may change from class to class or from instructor to instructor.

Moreover, algebra is capable of far more than only containing a multitude of situations in a single formula. Like Play-Doh, it can render variable expressions malleable enough to maneuver and transform them into radically different shapes and forms that generate novel insights. One especially pronounced instance of this capacity is visible in quadratic equations, where we can represent a galaxy of such equations with the single big algebra formula *ax*^{2} + *bx* + *c* = 0. From this we can apply the technique for solving each such individual equation, once and for all, to this single super-equation and derive the famous quadratic formula:

(See Appendix 1 for more details.) This formula holds the collective results of infinitely many individual acts in suspended animation, showing through the spectacular use of the parameters—*a*, *b*, and *c*—the final form that all of the specific instances can end up taking. We also saw the potential of big algebra in the various interpretations of the bill denominations problem from __Chapter 11__.

Significant scientific laws and processes have historically been captured by algebraic formulas capable of handling a wide array of different inputs—including mass, charge, the coefficient of friction, the spring constant, electrical resistivity, and electrical conductivity—through the use of parameters. Similarly, in real-life financial situations, parameters can stand in for fixed numerical information such as the sales tax in various towns, the interest rates at a given time, a principal amount invested, the price of gas per gallon on a given day, or the hourly wage of various workers. Although these constant quantities may be fixed in a particular location, for a certain period of time, or concerning a specific person, they can vary depending on the unique circumstances of the problem. In tuning our parameters from scenario to scenario, we become like patrons at the algebraic sports bar where—instead of changing channels between football games—we change channels from scenario to scenario identifying their similarities and differences, their constants and variables.

**A HIGHER CALLING**

As breathtaking as it may be at first sight, the Grand Canyon becomes all the more impressive once we understand its dimensions, what it contains, and how it was slowly carved out by the Colorado River over millions of years. The realization that some of the same smaller-scale processes at work in the erosion you witness in your backyard or at a local municipal park along the river also created this immense natural wonder is a spectacular and powerful grand confluence of ideas.

Emergent viewpoints in geology—such as understanding the relationship between micro and macro processes—makes much of what we observe on Earth more comprehensible and less mysterious. Moreover, these viewpoints provide even nonexperts, who have some understanding and awareness of them, an entire framework within which to place and analyze new and unfamiliar landscapes that they happen upon. Such awareness can also lead individuals to acquire a greater appreciation and awareness of their natural environment, which may ultimately lead them to become even less intimidated by the subject of geology itself. The same can be said for astronomical knowledge. As Bernard de Fontenelle stated, “Nature… is never so admirable, nor so admired as when it is known.”^{7}

Consider how we might apply the same principle to large swaths of basic numeration and even some aspects of geometry. Most people can tell you very little about the nuances of land surveying but, at the same time, are often quite comfortable with basic geometric concepts such as length, angles, area, and volume. Moreover, those same people can successfully apply those concepts to calculating the distance between two towns, the heights of people or structures, the square footage of a house or room, or the area of a garden.

How might we frame such a relationship between micro and macro algebra? What would it look like—and is it even possible—for someone with a rudimentary understanding of variable quantities, and a passing familiarity with algebra from a distant course or two, to extrapolate that basic proficiency to new situations involving variable phenomena? Variation and all that it entails is a much more abstract idea, evidenced by the fact that most people readily internalize elementary arithmetic, whereas conceptual retention of fundamental algebraic principles is orders of magnitude lower in the general population.

Algebra in a sense does for arithmetic what arithmetic does for the general notions of the size of a collection, measurement, and ordering. That is, algebra offers a set of rules for manipulating mathematical symbols that represent objects and operations from arithmetic, a subject that itself offers a set of rules for manipulating symbols that represent numerical quantities. A second level of symbolization and a second level of generalization can be tough to master, but symbols at both levels serve a powerful purpose.

One such power of symbolism is that symbols enable us to speak about things in their absence.__ ^{8}__ This is equally true of drawings, photographs, and maps. In many cases such as with military units, sports, or travel, such representations allow us to better understand the things being represented while probing them for weakness or limitations and gleaning new insights. So too is it with the alphabetic, diagrammatic, and graphical symbols we use in algebra.

As art education scholar Elliot Eisner wrote in *The Arts and the Creation of the Mind*, “Ideas and images are very difficult to hold onto unless they are inscribed in a material that gives them at least a kind of semipermanence.”__ ^{9}__ Yet, how can we present algebra in such a way that its symbols and methods illuminate and expand our mathematical knowledge, rather than obscure and obstruct it? This is one of the central problems in algebra education—and it has proven to be a tougher nut to crack than for arithmetic.

What do people remember about or take away from the other classes they take? Though undoubtedly not as much as their teachers might hope, students do perhaps take general lessons away with them into their future coursework and professional lives. They may even learn some of these lessons outside of class.

Take science classes, for instance. Former science students are likely to be at least somewhat familiar with the concepts of atoms, planets, stars, power, and forces, as well as physical phenomena they personally can experience like electricity, gravity, velocity, and acceleration. Some can also recall the names of a few of the famous scientists associated with major discoveries. From chemistry, even a student who has forgotten how to balance an equation is likely to know that mixing unknown substances together could produce dangerous gases or even cause explosive reactions. From biology, they know that tiny microorganisms can carry contagious diseases and that airborne pathogenic microbes can infect us.

What do people think about when they recall algebra? Common responses may involve letters, equations, and manipulations, but rarely with an appreciation of what those elements are really about. If students leaving algebra classrooms can’t articulate why algebra is conceptually significant, even at the most basic level, this indicates a breakdown in algebra education relative to other subjects in the secondary school or college curriculum—something that has made it a topic of ongoing discourse in math education. This book has been an attempt to contribute a few ideas to that discourse while simultaneously fostering a greater appreciation of the magic and wonder of algebra in general—one of many possible higher callings of algebra educators.

William Thurston once wrote that “it is easy to forget that mathematics is primarily a tool for human thought,”__ ^{10}__ and it’s true that the perspective that’s often lost in translation in the algebra classroom is that algebra can shape the way that we

Though teaching algebra as a way of thinking rather than as a means to an end may be difficult to achieve in the current educational environment, a student outcome that achieves a more holistic understanding of algebra—and emphasizes the purposes and meaning behind the procedures being taught—would be a step in the right direction and is a worthwhile goal in and of itself. I believe that Daniel Willingham is correct in his claim that, though many of the complaints against algebra involve questions of relevance outside of the classroom, the far greater concern is that many students simply don’t understand the rationale behind the calculations they are performing.__ ^{11}__ The unfortunate reality that most of them find no cohesion—no conceptual organizing principle either within the subject itself or in their efforts—doesn’t help matters either.

If students understood the purpose of algebra from the beginning, they may leave their studies with both a better feeling toward the subject and a better sense of how they might make use of what they’ve learned. This is by no means a trivial thing, amounting to what John Dewey calls having a worthwhile experience (as discussed in the introduction).__ ^{12}__ However, this is not a comprehensive picture of what may be possible. Students certainly will encounter quantitative variation elsewhere in life—what would be some of the things that they could think about when they encounter it?

One of the things we’ve touched on several times throughout the book is that numerical variation in everyday life is not always transparent, and that it can take some work to even be aware of its presence. When we encounter quantitative variation in the wild, it is often as a specific instance of a numerical value. Sometimes this particular value may be all that we are interested in, just like sometimes we only want to know the specific temperature and weather conditions on a certain day.

However, more is available if we want. Just as a person can learn much more about a geographical region by being aware of the range of temperatures and weather over a period of time—its climate—so too can the algebraic way of thinking help us to better understand the climate of a particular phenomenon. This can be done in part by having a better feel for the range of values it can take on; so, when we encounter a particular number for a particular type of behavior, it may be productive to look at that value as a particular instance—like a particular temperature—of something more general. Questions we could ask include: What type of algebraic climate could produce these values? What patterns can we identify that will tell us more about it?

Think of it this way. An alien coming to Earth might first see children, young adults, and the elderly as distinctly different species of people, not understanding the aging process or that one group morphs into another group over time. However, after much observation over time, they could eventually reach this conclusion on their own by studying the relationships and interactions between each group.

Encountering specific instances of variation is a bit like this. Modeling scenarios algebraically gives us the ability to tie together situations that we may not initially understand or see as connected. Alternatively, if a range of values are what is first encountered, then we can go the other way and ask if there is some core mechanism that ties them together in a formulaic way: Algebra’s investigative properties work both ways. As we discussed in __Chapter 9__, part of understanding that core mechanism may include trying to find out if it easily splits into scenario variables and regular variables and identifying what those are. This can be a useful organizing principle even when a detailed, technical understanding is not attainable.

As a mathematician educator, I’ve found that developing this kind of higher-order algebraic awareness tends to be easier for adult students. Adults are trying to weave much more complicated tapestries in their understanding than children who are still developing; this instinct can be leveraged and built upon through a humanistic approach to mathematics that connects with their own experiences. In my teaching, I aim to give my students a sense of the bigger picture of algebra’s contribution to our collective understanding, which can mean giving older students a productive new perspective that alters or evolves the way they see the world and what they already know about it. One of my goals in writing this book has been to capture the magic and impact that such a global shift in perspective has had for my adult students—and though there are many different ways to teach algebra and my approach may not necessarily be the most effective for the high-school classroom, my hope in this final movement of our algebraic symphony is that you will close this book with a better sense of what algebra has to offer us. I may not have convinced you that the algebraic way is “fun,” but I hope you’ll agree that algebra can be interesting, accessible, and full of possibilities.

Figuring out how to successfully incorporate topics into an algebraic setting can be likened to figuring out how to incorporate various ingredients into cooking a good meal. We know what needs to be added—some ingredients essential and others optional—but it is the proper mixing of these—along with proper amounts of heat and time—that turn our efforts into a good meal or not. As just one cook in a kitchen that features a wide assortment of recipes and cooking styles, I hope you have enjoyed my offerings.

**ALGEBRA THE BEAUTIFUL**

We’ve thought about algebra throughout this book as a vast continent of knowledge, containing both areas that we already know fairly well and tracts of less penetrated terrain brimming with unused potential for readers to explore. It is a grand subject whose ability to exploit a remarkable property of nature is key—that property being that it is possible to represent, describe, mimic, and transform a wide array of phenomena using symbolic systems that obey certain protocols.

Paradoxically, it is exactly these protocols that are often looked upon as being among the most monotonous of the features of algebra, no doubt contributing in part to the subject’s large PR problem. Yet, it is exactly these procedures—along with the fact that they were acquired to deal with one situation but are capable of being continually reused to deal with other situations—that account for some of the most intense aspects of the beauty of the subject. Imagine a retail store gift card with a zero balance. Such a card can be looked upon as being a valueless rectangular piece of plastic or as an object having a lot of potential. This potential can be realized through recharging or reloading the card by adding money to its balance. Once this has been done, a wide assortment of possible purchases emerges.

Similarly, the symbols and protocols of algebra can be looked upon as being completely devoid of value, the many manipulations just so much arbitrary, meaningless ritual. However, like the empty gift card, these protocols and rituals have the potential to be given immense value. Algebraic expressions and operational procedures can be continually “charged up” to represent and say important things about a wide array of quantitative variations, including the ability to take these charged-up representations and make spectacular transformations of form to find out unknown, new information and insights almost like no other discipline that preceded it or that isn’t presently underwritten by it. In other words, the apparently mindless ritual can be electrified to great purpose—acquiring immense worth. In this, it shares great similarities to computer science, natural science, and engineering.

Thus, elementary algebra when viewed from a certain perspective can contain in miniature some demonstration of how these other disciplines contribute to the advancement of human knowledge and our understanding of the universe around and beyond us. It is a subject with a fraught pedagogical history, at one time having its secrets completely masked from public view, like a half-mythical impenetrable forest, and yet at the present time having many of its secrets hidden from the public in plain sight, with most people unable to appreciate the algebraic forest now for its many trees.

Algebra is a heritage that belongs to every one of us—truly one of the intellectual wonders of the ancient and modern world, and though probably not perched as high, by discipline, as one of the top seven such wonders, it certainly has helped bankroll some of those in the top spots.

Regardless of where it sits, algebra is a vast, scenic, wide-ranging continent of possibilities that provides conceptual fuel for mathematics, as well as much of science, engineering, and a host of other disciplines vital to modern life. It was already a very big deal during its modern symbolic ascent back in the age of printing presses, Mercator projections, armadas, Kepler, and slide rules, and it still remains a big deal in this current era of the internet, global positioning systems, carrier battle groups, data science, and computers.

Algebra the Beautiful—fertile, electrical soil indeed.