7
To share in the delight and the intellectual experience of mathematics—to fly where before we walked—that is the goal of a mathematical education.
—William P. Thurston (1946–2012), “Mathematical Education,” Notices of the American Mathematical Society
Mathematics is the art of giving the same name to different things… When language has been well chosen, one is astonished to find that all demonstrations made for a known object apply immediately to many new objects: nothing requires to be changed, not even the terms, since the names have become the same.
—Henri Poincaré (1854–1912), Science and Method
The idea of requiring algebra for all remains under assault from many corners in the United States today. Burning like an eternal flame, still, is the centuries-old argument that most people won’t ever use the algebra that they learn in school anyway, so why make them take it at all?
Such thinking contrasts sharply with the heady days of the late nineteenth century when influential educators, no doubt in part after having observed the spectacular successes of injecting arithmetic earlier into and throughout the curriculum, wanted to see something similar happen on a wider scale with algebra. At least in the public high schools, that is, whose day in America had arrived.
Going back a hundred years further still to the 1700s, even something as simple and practical as basic written arithmetic was not in the public trust at all, generally being taught only to a relative handful of privileged students of adolescent age. This practice had its roots going all the way back to thirteenth-century medieval Italy with the introduction of the then revolutionary abbaco schools, which taught select students in this age group written arithmetic (among other subjects) for commerce—in the vernacular instead of Latin.^{1}
The idea of an organized public education for all was still in its infancy in the eighteenth century, with most young Americans not attending school at all. And for those who did, most ended their formal education long before their adolescence. Thus, most Americans knew very little arithmetic. Such was the state of math education in this country that arithmetic was, for a time, taught in the junior and senior years at Harvard College. It was only sometime around 1780 that the subject was moved down to the freshman year.^{2}^{,}^{3}
George Washington thought it an issue of patriotic concern. In his 1788 letter to Nicholas Pike, author of the first arithmetic textbook written by an American after the revolution (A New and Complete System of Arithmetic Composed for the Use of the Citizens of the United States), he states:
But I should do violence to my own feelings, if I suppressed an acknowledgement of the belief that the work itself is calculated to be equally useful and honorable to the United States.… The science of figures, to a certain degree, is not only indispensably requisite in every walk of civilised life, but the investigation of mathematical truths accustoms the mind to method and correctness in reason.… From the high ground of Mathematical and Philosophical demonstration, we are insensibly led to far nobler speculations and sublime meditations.^{4}
Elsewhere in the world at the time—notably in Switzerland—remarkable educational reforms were being proposed and developed. And one of the most significant curricular innovations to emerge from it all was to be elementary arithmetic. The reforms in its teaching and presentation would be of such a force, excitement, and extent that important parts of the subject would soon storm their way spectacularly down into the public elementary school—eventually coming to rival even reading, vocabulary, writing, and spelling, themselves, in importance.
Arithmetic’s stunning migration down to the earliest years of the public grade school (wresting it out of the hands of the privileged and the college-educated, thus making it potentially accessible and understandable to nearly everyone), though not perfect, still remains one of the most successful and enduring educational reforms of all time. We are all among the beneficiaries of this late eighteenth- and early nineteenth-century reform in arithmetic, a reform so successful that few still seriously question the subject’s place in education. It has become one of the pillars in elementary instruction.
Nothing close to similar has happened, unfortunately, with the introduction of algebra into the American public school curriculum. The educational reform cementing its near universal comprehension, acceptance, and appreciation—though highly desired by many in the late nineteenth century—still has yet to occur.
Why is this? An enduring educational puzzle? Or does algebra just have a PR problem?
LOFTY, ILLUSORY GOALS
The goals for the inclusion of algebra in the American public school curriculum have always been lofty and somewhat elusive. The subject in many ways took a parallel trajectory to that of basic arithmetic, as both were commonly taught at the college level early on before being pushed down a level or two in the curriculum. Its retention, along with geometry, as part of the standard college course of study was vigorously defended and leveraged in the highly influential “Yale Report of 1828.”^{5}
The push for injecting algebra into the secondary school curriculum can be seen in the gradual incorporation of the subject in some of the private academy (private secondary or high school) textbooks of the eighteenth and early nineteenth centuries.^{6} Perhaps this was done initially to give the more advanced students an advantage before college—as AP courses do today—but by 1820, Harvard had made it a requirement for admission, spectacularly going in just over four decades from teaching its students arithmetic in the upper class years to now requiring that they knew not only arithmetic but also some algebra before even being admitted to the institution.^{7}
The relatively small but growing number of public secondary schools also began to incorporate algebra into their curricula, and by 1850 its place in the high school was no longer out of the ordinary.^{8} The inclusion of other less traditional school courses for the time (history, geology, art, music, bookkeeping, surveying, physical education, modern languages, training for future teachers, modern literature, geography, chemistry, and manual training) followed suit, leading to a veritable explosion in the diversity of curricula around the country.^{9} By the latter part of the century, inconsistent course offerings from school to school and competing philosophies about the goals and purposes of secondary education—indeed all of public education—were becoming a burden for colleges and universities to decide on what to accept for admission.
And even more strongly in turn, the diversity in admissions requirements and course offerings from the many colleges and universities along with an increasingly diverse student body were beginning to overwhelm some high schools on which subjects and prerequisites (if any) they should fit their curricula to, because it was becoming more and more difficult to satisfy the needs of everyone.^{10}
Noted educational historian Diane Ravitch summarizes the issues of the day nicely:
In an age marked by the development of systems and organization, the schools seemed helter-skelter, lacking uniformity or standards. What should be taught? To whom? At what age? For how long? What were the best methods? What subjects should be required for college entrance? Should “modern” subjects such as history and science be accepted for college admission? Should students be admitted to college who had not studied the ancient languages? Should there be different treatment, even different curricula, for the great majority of students who were not college-bound? Should high schools offer manual training and commercial subjects?^{11}
In a sweeping effort to provide some professional guidance, the National Education Association (NEA) in July of 1892 formed a committee of ten of the most influential educators in the country to study the problem and lead the charge to create a set of standards and guidelines for the American high school curriculum.^{12}
Though the entire project was officially named the Committee on Secondary School Studies, the collective effort came to be better known as the famous “Report of the Committee of Ten”—so named for the ten-man committee that facilitated the project and composed the final report. However, in compiling the final report, the committee was advised by nine subcommittees (each with ten members) to ensure that their recommendations were informed by subject-matter experts and educators, one for each of the following subject areas^{13}:
1. Latin
2. Greek
3. English
4. Other Modern Languages
5. Mathematics
6. Physics, Astronomy, and Chemistry
7. Natural History (Biology, including Botany, Zoology, and Physiology)
8. History, Civil Government, and Political Economy
9. Geography (Physical Geography, Geology, and Meteorology)
It is worth pointing out that, at this time, higher education itself was also in a state of tremendous flux, with the idea of the classical college and its prescribed curriculum as endorsed by the “Yale Report” now giving way to that of the modern university with its elective curriculum and postgraduate and professional schools. Thus, the Committee of Ten and its subcommittee members—47 from colleges/universities, 42 from schools, and one a government official who formerly worked at a university—were in the vanguard of these seismic changes and among the most reform-minded educators in the country.^{14}
The subcommittees each met in three-day-long conferences in late December 1892. NEA Board Chairman (and noted education author) Norman A. Calkins had attempted to negotiate with the railroad companies to get reduced fares for conference attendees who were traveling from afar, but though able to procure some reductions, he was not as successful as had been hoped.^{15} The conference locations and dates were as follows^{16}:
Conference |
Host Institution |
Location |
Dates |
Latin |
University of Michigan |
Ann Arbor, MI |
December 28–30, 1892 |
Greek |
University of Michigan |
Ann Arbor, MI |
December 28–30, 1892 |
English |
Vassar College |
Poughkeepsie, NY |
December 28–30, 1892 |
Other Modern Languages |
Bureau of Education |
Washington, DC |
December 28–30, 1892 |
Mathematics |
Harvard University |
Cambridge, MA |
December 28–30, 1892 |
Physics, Astronomy, and Chemistry |
University of Chicago |
Chicago, IL |
December 28–30, 1892 |
Natural History (Biology including Botany, Zoology, and Physiology) |
University of Chicago |
Chicago, IL |
December 28–30, 1892 |
History, Civil Government, and Political Economy |
University of Wisconsin |
Madison, WI |
December 28–30, 1892 |
Geography (Physical Geography, Geology, and Meteorology) |
Cook County Normal School |
Englewood, IL |
December 28–30, 1892 |
The discussions at the meetings reportedly were “frank, earnest, and thorough”; yet in the end, they resulted in remarkable agreement amongst the various groups (with only two subcommittees submitting minority reports). The final reports from all of the conferences were completed by the middle of July 1893.^{17}
Each of these reports were delivered to the Committee of Ten, and by December 1893, they were ready to share their final consolidated report. This report is still, over 125 years later, one of the most influential educational documents ever issued in the United States.
Some of the harshest criticism against requiring algebra, among other courses, in the high school curriculum today is often directly or implicitly aimed at the report’s conclusions and the individual members of the Committee of Ten. Unfortunately, much of it, according to Ravitch, David Angus, Jeffrey Mirel, and other historians of education, is based on myths, half-truths, and outright mischaracterizations on the committee’s aims—as well as on misconceptions of the state of American high school education at the turn of the century.^{18} Much of it also seems to ignore the fact that the case against algebra isn’t new. Powerful, detailed, sustained, and successful crusades have already been waged in the past against requiring algebra and other academic subjects in the secondary curriculum—especially during the first part of the twentieth century—with highly problematic long-term effects.^{19}
On the eve of Pearl Harbor in November of 1941, Admiral Chester Nimitz himself, in a widely circulated letter, expressed alarm about what he saw as the decreasing level of mathematical education in naval officer candidates. Many educators—naval and civilian—attributed this decline to the more than 25-year assault on basic mathematics instruction that had been waged by prominent leaders in the progressive education movement.^{20} The alarm bells continued to sound after the war, most prominently at the University of Illinois in the early 1950s, leading to an influential reform movement, whose math component was initially spearheaded by Max Beberman, Gertrude Hendrix, and Herbert Vaughan, and that ultimately culminated in the New Math of the Sputnik era.
Even in their day, the views of the committee were controversial. Committee Chairman and Harvard president, Charles Eliot, who had spearheaded curricular reform through pushing for the elective system over the traditional prescribed curriculum in American colleges, stated in 1892:
It is a curious fact that we Americans habitually underestimate the capacity of pupils at almost every stage of education, from the primary school through the university. The expectation of attainment for the American child, or for the American college student, is much lower than the expectation of attainment for the European. This error has been very grave in its effects on American education…^{21}
Florian Cajori, a prominent member of the math subcommittee, shared similar sentiments, stating two years earlier in 1890:
One of the most baneful delusions by which the minds, not only of students, but even of many teachers of mathematics in our classical colleges have been afflicted is, that mathematics can be mastered by the favored few, but lies beyond the grasp and power of the ordinary mind. This chimera has worked an untold amount of mischief in mathematical education.… This humiliating opinion of the powers of the average human mind is one of the most unfortunate delusions which have ever misled the minds of American students and educators. It has prevailed among us from the earliest times.^{22}
Such pronouncements give some indication of why the committee wanted to keep subjects like algebra in the high school curriculum.
Critics argue that the Committee of Ten members—mostly of privileged background—were out of touch with the needs of the new crop of students entering high school, and that their elitist outlook biased them toward a view of public high school solely as preparation for college-bound students with no thought or concern for students whose education would end with secondary school.^{23} However, the 1893 report itself states:
The secondary schools of the United States, taken as a whole, do not exist for the purpose of preparing boys and girls for colleges. Only an insignificant percentage of the graduates of these schools go to colleges or scientific schools. Their main function is to prepare for the duties of life that small proportion of all the children in the country—a proportion small in number, but very important to the welfare of the nation—who show themselves able to profit by an education prolonged to the eighteenth year, and whose parents are able to support them while they remain so long at school.… A secondary school programme intended for national use must therefore be made for those children whose education is not to be pursued beyond the secondary school. The preparation of a few pupils for college or scientific school should in the ordinary secondary school be the incidental, and not the principal object.^{24}
This is an extremely ambitious program for the secondary school, then sometimes called the “People’s College.” At its root were two beliefs: (1) A broad-based education in some combination of the nine subject areas was the best way to develop a well-informed and reasoned American individual, and (2) all high school students, college-bound or not, should have equal access to such studies once considered open only to the most privileged of youths. This course of study was deemed by committee members to be important for participatory citizenship and leadership in the democratic United States, regardless of eventual employment destination.^{25}
Consider literacy, and how reading is seen as a vital core skill required to navigate in the modern world, no matter what else a student may learn or do in the future. To the Committee of Ten, certain parts in the nine subject areas were also important for a citizen, of the time, receiving a high school experience or certification to be proficient in—not college as an end in itself, but a solid broad exposure to what they viewed as being among the central and rhythmic strands of knowledge. This was most especially true for those students who would not continue beyond high school, because committee members felt that (for these students) this exposure in secondary school would be their only chance to ever explore these strands in a systematic way before moving on to receive more job-specific training in their chosen occupations. Afterward, college could also be at the ready if some of these students later changed their mind about continuing their education. This approximates an essential tenet in the majority viewpoint of the Committee of Ten.
Even by 1893, the arguments for continuing to accord classical languages like Latin and Greek such prominence in the curriculum were steadily losing ground, as a prime justification for them, as sharpeners of the mind—the mental discipline theory—increasingly came under attack. Additionally, as more comparatively recent literature such as Shakespeare’s works and the works of American authors gained increasing acceptance into the curriculum, and as translations of works in other languages became more available, many believed there was a decreased need for the ancient languages to be required any longer for the reading of ancient literature. Thus, it became ever more difficult to continue selling the idea that the study of these languages offered any marked advantage over the study of modern languages. Evidence of this can be seen in the steady decline in prominence of Greek and later Latin in the secondary curriculum, which continued on into the twentieth century.
The seven other subject areas had a more robust array of defenses to bolster up their claims as to why they should be a part of the core strands of general knowledge.
William Torrey Harris, US Commissioner of Education and a member of the Committee of Ten, had stated many years earlier that a standardized curriculum for public education was necessary in order to avoid creating a caste system in America, where such education would be reserved for students of privilege in private academies and not available to students who attended public schools.^{26} He viewed such a system as being fundamentally nondemocratic.^{27}
Others didn’t think it possible nor desirable to achieve these aims with public secondary education for all. One of the harshest opponents of the Committee’s recommendations was G. Stanley Hall, president of Clark University in Massachusetts and an internationally respected psychologist. Hall later criticized the Committee’s conclusions because he thought that their aims were too idealistic and that statements such as those made by contributors to the final report do not “apply to the great army of incapables, shading down to those who should be in schools for dullards or subnormal children, for whose mental development heredity decrees a slow pace and early arrest, and for whom by general consent both studies and methods must be different.”^{28} Hall’s strongly worded opinion, encapsulated in this one hyperbolic statement, has for over a century kept his name at the forefront of those cited as being in disagreement with the Committee’s conclusions.
Many felt similarly to Hall regarding the capacity (or lack thereof) of the average American student—even those who would have chosen to use less condescending language—arguing for less standardization and for the creation of significantly more varied educational tracks for high school students of “varying ability and interests.” Still more conservative commentators felt that the Committee had already been far too generous in its reforms by proposing to include “newer” subjects such as biology, geography, and history and stressed that the classical, prescribed curriculum as recertified in 1828 with Latin and Greek at its core should be shored up in the high schools and adhered to—especially as the battle to retain the classical curriculum at the post-secondary level was steadily being lost.^{29}
The mathematics subcommittee tried to strike a balance in its recommendations. The group recognized the enormous difficulties many students faced under the existing systems of instruction, and realized that substantial change was necessary to deal with them, arguing for the need to create a more integrated understanding and appreciation of mathematics in the student. To do this, they suggested removing the especially arduous or arcane problems that were found in some of the textbooks of the day, as well as replacing some of the more demanding and opaque arithmetic procedures with the simpler, more unifying algebraic methods.^{30} These latter recommended revisions took firm root in the 1900s.
The math subcommittee also campaigned for a different type of in-class instruction, one that didn’t rely on rote memorization alone, but rather greatly enhanced the powers of memory by motivating the subject first through the use of more concrete examples.^{31} Additionally, they wanted to see instructors become more effective at explaining why mathematics works and how to apply it, believing that doing so would make the subject more understandable, broaden its appeal, and better accommodate the tidal wave of new students coming into public high schools.
Comparing nineteenth-century arithmetic and algebra textbooks with those from the twentieth century, we can see that implementation of some of these ideas did occur and has resulted in some positive progress.^{32} However, if these arguments for reform still sound familiar today—125+ years and running—it may be because the subcommittee’s lofty goals to make math more accessible and appealing haven’t yet been fully realized. Most contemporary educational narratives—in the press and elsewhere—don’t indicate disagreement.
Though the members of the controlling Committee of Ten in their final report were a bit firmer in their recommendations for the math course of study, the ten members of the math subcommittee had allowed for diversification beyond a certain point. But for them, that diversification (e.g., commercial mathematics and bookkeeping) should happen after the first course in algebra, not before.^{33} A number of the arguments today passionately contend that this diversification should happen before the first course in algebra, not after.
The following table shows for a given date the percentages of students in the last four years of high school enrolled in algebra, geometry, and trigonometry^{34}:
School Year |
Algebra |
Geometry |
Trigonometry |
1890 |
45.4% |
21.3% |
1.9% |
1900 |
56.3% |
27.4% |
1.9% |
1910 |
56.9% |
30.9% |
1.9% |
1915 |
48.8% |
26.5% |
1.5% |
1922 |
40.2% |
22.7% |
1.5% |
1928 |
35.2% |
19.8% |
1.3% |
1934 |
30.4% |
17.1% |
1.3% |
1949 |
26.8% |
12.8% |
2.0% |
1952–1953 |
24.6% |
11.6% |
1.7% |
1956–1957 |
28.7% |
13.6% |
2.9% |
For the snapshot taken in 1890, 45.4% of all students then enrolled in the 9th, 10th, 11th, and 12th grades were taking a course in algebra during that year.
ALGEBRA FOR WHOM?
Whatever the argument, the importance of algebra to the existence and continued expansion of critical sectors in our modern high-tech society is without question. This is even more true now than it was in 1893. For starters, many new technologies and modern conveniences in transportation, communications, energy, electronic computation, military science, and consumer products—that we’ve come to enjoy over the last 125 years—owe their existence to advancements in science and technology, fields that critically depend on mathematics.
So, someone needs to learn algebra. Few, if any, dispute this. But should everyone be required to learn some?
Clearly, those students who pursue advanced studies in a STEM—science, technology, engineering, and mathematics—field will use the elementary algebra they learn in school. But what about students whose sole exposure to the subject will be a single course or two? Can they really take away anything of lasting substance and value from the experience—other than way more frustration and grief than most of them should endure?
Similar questions, regarding more advanced courses, can be raised by STEM students in higher education. These students, particularly engineers, are required to take the calculus series (and often a math class or two beyond), and a lot of them struggle mightily in these classes, too. Later on, many discover, especially with the tools of modern technology, that they rarely use this higher-level math professionally in the form it was taught to them in college. So, they might just as easily ask: What exactly are they taking away from these more advanced math courses?
Yet, most engineers don’t seriously ask this question. Why is this? Whether or not the more advanced math courses have a direct practical application in their working life, most engineers know that these courses give them powerful and useful conceptual fuel. We can see this reasoning manifest itself in several ways. For one, from a strictly pragmatic standpoint, most engineers know that more advanced mathematical knowledge qualifies them for a broader pool of jobs in their field. This is not an insignificant factor, as many people end up having several different jobs over the course of their lives—and as Stella Adler reminds us, the more comprehensive our education, the better our ability to perform in a variety of roles.
Another way, perhaps, to think about this is in terms of active and passive vocabulary. A person’s active vocabulary consists of the words that they regularly use in their thinking, writing, and speaking, whereas their passive vocabulary includes all of the words they recognize and understand.^{35} If we apply the logic that is often leveled at the algebra requirement toward our vocabularies, then a person shouldn’t have a passive vocabulary at all. After all, why learn words that you aren’t going to regularly use?
But clearly our passive vocabularies are important too, as they enable us to understand and appreciate a wider range of concepts than we can ourselves directly communicate. We can, in a short span of time, read a book that may have taken a seasoned expert over a decade of concentrated effort and research to write, thus becoming an active participant in a sophisticated intellectual activity that we could in no way have produced ourselves. Our passive vocabularies grant us the ability to more effectively dial in and interpret parts of the greater world outside of our own experience. This is even more apparent in young children who can create far less in language than they are able to understand and act upon—and what they hear during those critical early years of life often makes a decisive difference in the rest of their lives.
This leads into our second point. Engineering students also bring a mathematical awareness into their professions from their coursework that allows them to comfortably understand much more technical material than they regularly use or ever need produce. This awareness serves, in a way, as part of their passive technical vocabularies, without which most would not be able to perform their jobs with confidence. Mathematicians sometimes liken this mathematical awareness to a type of maturity—often calling it “mathematical maturity.” Both points still retain some water if we bring them back to the level of elementary algebra, where we will call the idea “algebraic awareness.”
This leads us into the chorus of this chapter and the next several chapters, the goal of which is to showcase a few of the recurring narratives and enhanced outlooks from which we can benefit by developing and tuning into this awareness.
THE STAR-SPANGLED BANNER
On a given summer day in the United States, our national anthem is sung in dozens of ballparks and sporting venues across the nation. Although each performance may be sung by different individuals, each with a different take and sound, we do not balk for a single moment in declaring that they are all singing the same song—“The Star-Spangled Banner.”
What is it about each unique performance that causes us no strain in identifying them all as being the same piece of music? Clearly, the fact that we recognize the same lyrics and tune is what binds all of the renditions together for us, making us look on each of them as simply different versions or interpretations of the same song.
A few members of this mighty ensemble are listed next:
This visual—of explicitly stated common structure (the lyrics) amidst all of the variation (the different performances)—has parallels all over the mathematical landscape and its application, providing us with potent conceptual fuel.
ALGEBRAIC SONGS
Consider the following five problems:
1. An unknown number added to ten more than fifteen times itself gives one hundred six. Find the number.
2. A 106-foot length of rope is cut into three pieces. The second piece is 10 feet longer than seven times the length of the first, and the third piece is eight times the length of the first piece; find the lengths of all three pieces.
3. Given a rectangle of perimeter twice 53 meters and whose length is five more than seven times its width, find its length and width.
4. If a power tool rents for $16 a day plus a one-time $10 processing fee, how many days can you rent the tool if you have four $20 bills, two tens, a five, and a dollar to spend?
5. Next Saturday, Barbara will be doing a job for a client that pays her $36 an hour. The job requires specialized computer services that cost $20 an hour to use in addition to a $40 setup fee. When she arrives on location for the job, she notices a $50 bill that the client left to say thanks for coming in on the weekend. How many hours does she need to work so that her total profit (including her tip) for the day is $106?
These problems are all about different things on the surface, but setting each of them up according to the techniques discussed in Chapter 5 shows that their algebraic essence is quite similar (see Appendix 2 for their analyses). In each case, simplification ultimately yields the equation 16x + 10 = 106:
Just as we can, through the lyrics, identify the many performances of “The Star-Spangled Banner” as being different versions of the same song, so too can we (in a way) think about the quantitative portions of each of these problems as being different versions of the same algebraic song identified through the algebraic expression (algebraic lyrics) 16x + 10 = 106.
This hints at what French mathematician and scholar Henri Poincaré meant when he said, “mathematics is the art of giving the same name to different things” (where “name” is “equation” in this case).^{36}
The algebraic ensemble is infinite in scope and undoubtedly includes many more possibilities, including real-world scenarios such as those in problems 4 and 5. Here, problems 4 and 5 correspond to cases where classroom word problems and real-world situations can overlap.
The basic algebra that we already know allows us to quickly solve all of these problems in an identical and familiar way:
The solution then is 6, which interprets in each of the respective problems as follows:
1. The number 6.
2. The first piece of rope is 6 feet long, which cascades to 52 feet and 48 feet for lengths of the second and third pieces, respectively.
3. The rectangle’s width is 6 meters, which cascades to 47 feet for its length.
4. 6 days.
5. 6 hours.
As Poincaré implies, the demonstration yields a solution that accurately applies to all of the various cases.
The symbolic tying together of a host of distinct objects and actions is not really a new concept, being one of the hallmarks of language itself. For example, the English word chair symbolically ties together millions of distinct objects of different sizes, weights, colors, and shapes. And the statement “the man walked down the road” ties together multiple trillions of distinct actions—any man (of any age, size, personality, etc.) walking down any road, for any length of time, on any given date (past or present) anywhere on the planet. And so on with other words and statements in English or in any other language. It is one of the things that gives language its great reach from a comparatively small glossary.
A huge difference, though, is that in the case of language most of us can naturally make the connection between a huge majority—if not all—of the objects we call chairs in spite of their diversity, and similarly with the different performances of “The Star-Spangled Banner.” Connecting them through the lyrics is almost as natural as recognizing a familiar voice or face.
In the case of algebra, however, the connections are more cloaked and can rarely be made from sensory inspection only. Moreover, algebraic similarities exist on different levels—ranging from the exact algebraic connections (that simplify to the same equation) made between problems 1 through 5—as is the case in this section—to the slightly more general points of contact (that represent the same type of equation) made between the problems in Chapter 5.
Sometimes it is only through mathematics that we can see how intimately connected diverse problems are. This is one of the most beautiful things about the subject. The cloaking of connections continues and can become increasingly obscure and sophisticated as the subject advances, and math’s ability to unearth even these similarities continues to inspire admiration and awe in those who bear witness to it.
A HEIGHTENED AWARENESS
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.^{37}
Mathematics is a continuum, so its ability to connect many things together shows itself from the start, even in elementary arithmetic. For example, the simple operation 32 × 48 can answer questions such as
• the area of a rectangle with a width of 32 feet and a length of 48 feet,
• the number of chairs in an auditorium that has 48 rows each with 32 chairs,
• the amount of money earned working for 32 hours if the wage is $48 an hour,
• the distance a train travels for 32 hours if its average speed is 48 miles per hour.
But in each of these cases, the mathematical connection through multiplication is almost as straightforward to recognize as it is recognizing that the different performances of “The Star-Spangled Banner” all represent the same song.
It is in algebra where things start to regularly get intricate and complicated enough to the point where the mathematical connections between phenomena become less obvious and more hidden, requiring a different form of treatment.
This makes the subject uniquely positioned to give students a powerful taste of what the higher aspirations of mathematics and science are truly about—other things like reorganization, unification, systematization, experimentation, and discovery, in addition to numerical computation and symbolic manipulation. And because these higher aspirations are central to the continuity and operation of our modern technological and data-driven society, the opportunity for more people to experience this taste should not be easily dismissed or caricaturized.
Many years ago, Alfred North Whitehead spoke to the issues of requiring courses, such as the classics, in an increasingly crowded curriculum:
We must remember that the whole problem of intellectual education is controlled by lack of time. If Methuselah was not a well-educated man, it was his own fault or that of his teachers. But our task is to deal with five years of secondary-school life. [A course in the] classics can only be defended on the ground that within that period, and sharing that period with other subjects, it can produce a necessary enrichment of intellectual character more quickly than any alternative discipline directed to the same object.^{38}
I believe that requiring algebra in the curriculum can also be defended on such elevated grounds. Its role in underwriting so much in our high-tech world, along with its ability to offer the proper mix of generality and concreteness, places it in a potentially commanding position of advancing powerful perspectives and insights across the disciplines—but only when utilized in its more optimal and expressive forms. The subject in education is not unlike a sophisticated airplane, which is capable of truly remarkable feats when operated at the proper speeds and altitudes, yet when flown improperly is also capable of stalling out, breaking apart, and suffering catastrophic failure.
The ability to see the impact of algebra—from taking only a course or two—on the wider world more often than not comes from a heightened awareness and comfort level with the subject. Though those who have had bad experiences with algebra may disagree, some of this heightened awareness and comfort may have already developed without them even noticing.
Consider the following: Last year there were 2,142,341 people who got married in the United States. How is this possible? Namely, how is it possible to have an odd number of people get married when there are two people involved in every marriage? It would seem that the number should always be even.
There are many ways to explain how this can happen—for instance, consider A, B, and C and imagine that A and B get married in January and then divorced in June. If B and C then get married in November while A remains single, we see that three people (an odd number) got married in a year.
Now, if you were able to follow this explanation without asking such questions as “How can three letters in the alphabet get married?” or “What in the world does an argument involving the first three letters of the alphabet have to do with 2,142,341 people getting married?”, then you may have already taken something algebraic away with you from your math classes.
You most likely have taken away a certain comfort level, or fluency, with the notion of letters acting as placeholders for varying information like numbers or individuals. If you were able to understand how the marriages between A, B, and C might explain a number that represents millions of marriages, then you also likely have developed some familiarity with generalizing quantitative relationships. That is, this simple example exposes the flaw in the assumption that a single individual can’t be involved in two different marriages in one year. From this, we realize that this circumstance could happen multiple times in a group numbering in the millions, allowing for the possibility of an odd number of people getting married.
To put it another way, we could also explain our resulting odd number if we simply said that someone could have gotten married, then got divorced, and then married somebody else in the same year while their first spouse remained unmarried. This reasoning involves unknowns and generalizations just as our algebraic-like rendering did—check to see how it relates to our initial framing involving the letters A, B, and C (and what is explicit and what is implied in the reasoning).
Heightened awareness exists in areas other than mathematics. For instance, say we have two people, one comfortable using the internet and the other not, who want to learn how to fix a flat tire on a car. The person who prefers traditional research only can certainly avail themselves of library resources, or try to find someone who can show them how to fix a flat tire. The other individual who is comfortable with the internet has these options plus the ability to do Google and YouTube searches, where they can find a variety of videos that discuss the process or show tires being changed on multiple types of cars. The latter person thus has more resources at their disposal and therefore a much better chance of successfully achieving their objective.
Consider a second example of two different people, one who has only basic internet literacy and the other who is more computer savvy. Say they both buy MP3 music players and want to transfer all of the stored music on their computers to these portable players, only to discover that the stored music is in a format—say RMJ, for Real Player software—that an MP3 player won’t play. The computer-knowledgeable person who knows that files can be converted from one format to another—for instance, Word files and Excel files to PDFs, or Photoshop files to JPEGs—decides to search for a program that can convert RMJ files to MP3s. A Google search quickly reveals that such programs exist, and 20 minutes and $10 later, they are happily converting their files to MP3 files. This person’s experience has provided a wider framework for them to situate a new problem and try to find a solution. The fact that this option even exists may never occur to the other person, and even if it does, they may lack the confidence and know-how to even download the files, much less carry out the actual conversion. They simply lack a broader contextual framework to resolve these types of situations.
These sorts of scenarios also occur when mathematical situations make planned and unplanned appearances in daily life, and people’s lack of confidence in the subject intimidates them into unnecessary avoidance and inaction. In other circumstances, simple mathematical interactions may be present yet remain unrecognized, along with basic operations and interpretations that could clarify the issues. In either case, with a bit more confidence and overall algebraic awareness, the mathematically disinclined could realize that they have additional resources at their disposal to simplify a problem they’re facing or learn more about it.
CONCLUSION
We have undertaken a very brief history of issues surrounding algebra’s place in the secondary school curriculum, and in so doing, we considered one of its central tensions: the value of elementary algebra to a contemporary high school or adult student. Here, we have argued that even if a student’s only exposure to the subject is a couple of classes in high school or college, algebra can be taught in such a way that creates a heightened awareness that will serve students in their future careers, whether in a STEM field or otherwise.
Next, we will explore this thesis in a bit more detail to see if we can really use any of the ideas from a course or two in elementary algebra in nontrivial ways to better understand and tune in to situations involving numerical variations that we may actually encounter—or hear about—in the course of our daily lives.