Solve for Why

The equals sign was an expression of equivalency. Now it’s an “answers button” for questions made in error

I lament the loss of play. I lament adulthood because it colonized my childhood imagination with facts and truths generated by the Captain Hooks of the past eager to indoctrinate a new soul into the soulcrusher that we call the workplace. I lament that drifting in the wind has become an action of deep anxiety — a journey with no destination that forces me to create plans and schedules to anchor myself within it. I lament that my actions no longer feel like my own but like a fulfillment of a mission put in place by a construct, safeguarded by people who start their sentences: “Technically…”

I have fallen out with friends. And with time I resent them more than I miss them. Does that happen to you? I decide they were wrong and I was right, and as time passes I decide they were wronger and I was righter, until soon they have become cartoonish villains bent on the destruction of society and I have become the humble martyr slogging toward a more just world.

It’s pathetic. It’s childish. It’s gonna be weird when I’m a parent, won’t it?

When I was 11 I flew to Sweden to visit my family and I packed two bags. One with clothes and toiletries, the other with two full binders of basketball cards and a calculator. Basketball cards list statistics on the back: offensive rebounds and turnovers per game, three-point shot percentage, and a smorgasbord of others to play with, and the calculator was my tool of efficient play. Any time I could get a hold of some numbers, I was excited to plug them into my calculator and find patterns. And it worked. I loved math. I love math. And now I teach math to high schoolers.

When calculators were first introduced into American classrooms in the early ’70s, one of the major arguments for their inclusion was that they would make math more fun. In 1975, a survey of math teachers for the Mathematics Teacher journal found that 96 percent of teachers thought that the “availability of calculators will permit treatment of more realistic application of mathematics, thus increasing student motivation.”

The notion that the only thing stopping students from having fun in math class is that they don’t see its usefulness is pervasive in our educational institutions. We believe that school has become a place for building workers, not learners, and therefore the classroom must be built around the idea of usefulness — for the worker. Douglas Lapp, a science curriculum specialist and avid calculator advocate in the ’70s, suggested that “The fundamental problem in math education still is that kids too often don’t know the meaning of mathematical education.” Jill Horlick, an elementary school math specialist around the same time who also advocated for calculators in classrooms, offered: “Kids normally think about the universe; they love to manipulate large numbers because it makes them feel important. Why stop [the child] from thinking beyond those numbers just because he doesn’t have the tools yet?”

When I struggle to find a “reason” my students should find the roots of a quadratic equation, colleagues say, “Tell them you’ve thrown their iPhone up in the air…” They are more adept at crafting answers than questions

These two seemingly compatible opinions point to the disconnect between the educator and the student. Educators see that students are interested in the big questions, the questions without final answers, and their response is to trick them into thinking they are doing something important. In the same article that quoted these two math educators, the calculator was heralded as a beacon of excitement in the classroom because it could offer students thrilling opportunities like “find[ing] how many cubic centimeters it would take to fill this room,” and how far the moon is in inches.

It feels impossible to ask these questions without putting on your most condescending voice and infusing it with false enthusiasm. As a math teacher I often find that this is the solution I’m given by administrators and colleagues when I’m struggling to find a “reason” to find the roots of a quadratic equation: “Tell them you’ve thrown their iPhone up in the air and they have to find out when it is going to come back down so that they can catch it and it doesn’t break into a million pieces.” They are more adept at crafting answers than questions, certainly; they don’t see that pretending these meaningless questions have meaning in order to serve as reason to attain an answer only forms distrust between the students and the adult in the classroom. I can hear the teenager in me grumbling: “If you think finding the cubic centimeters in this room is important, that it’s worth my time to calculate when the express train will catch up to the local train, why would I trust you to tell me what’s important.”

We are not the first society to create word problems in order to “make math fun.” The text that is often referred to as the origin of Chinese mathematics was a collection of word problems compiled over eight centuries and finished around 186 BCE: The Nine Chapters of the Mathematical Art was a series of parables that read alarmingly like questions we now see on the SAT. Questions like: “A square town has a gate at the midpoint of each side. Twenty paces north of the north gate there is a tree which is visible from a point reached by walking from the south gate 14 paces south and then 1775 paces west. Find the side of the square.”

Soon after the creation of The Nine Chapters, the Han Dynasty began the first massive public school initiative as well as competing private-school formats that were sanctioned by the emperor. Dong Zhongshu, an influential thought leader in the beginning of the Han Dynasty, wrote: “To wish for worthy men without having nurtured one’s corps of officers is like looking for beautiful patterns in unpolished jade. In nurturing officers, nothing could be more important than establishing an academy. An academy is the gateway of worthy gentlemen and the source of the transformations of the proper teaching.” He went on: “I would wish your majesty to erect a central academy and appoint enlightened teachers to it in order to nurture the gentlemen of the world. They should be periodically tested in order to push them to the limits of their abilities. If this is done, it should be possible to recruit the flower of the empire’s youth.”

From 124 BCE to 146 CE the number of boshi disciples (analogous to university students) rose from 50 to 30,000. The Nine Chapters became a central text in the academy. And therefore these word problems became central in the selection of who would take power in and around the new government.

By 600 CE, the Sui Dynasty created a civil service exam meant to determine a meritocracy of hiring through the schools. Those who did well in school got better jobs. The Song Dynasty (960-1279) created an academia to push stellar students into research and writing. This fostered a group of scholar-gentry civil servants (an intellectual class). But all the while, this intellectual class was getting corrupted by the test that put them in place. As Robert Eno, professor of early Chinese History and Thought at Indiana University Bloomington, states, “A small percentage of brilliant or outstanding ministers and scholars did help to mitigate the brutality of China’s autocratic political culture. But a far greater number of academy or examination graduates ended up either cynics or blatant hypocrites, and while they mouthed Confucian pieties to enhance their status and coerce the population they ruled, their personal and political conduct was deeply scarred with loyalties to self and family over the state and the people.”

The pendulum swung back. As we’ve seen with any meritocracy there are people left out, and the upper class often considers those people strong but stupid. The Yuan Dynasty (1271-1368), a militaristic and colonial power, took control of the educational system and began using it as an ideological education. They separated people they felt might unite and rebel and created a caste system that castigated the formerly elite Han and Nanren people to the bottom two social classes, which eventually came with restricted weapon and pet ownership as well as stricter punishment for similar crimes. The corruption of a meritocracy is always based on who gives the merit, and when people feel left out of the elite it causes a militaristic uprising that eventually takes control of the meritocracy and finds new, often more nationalist, ways to judge.

When I was in fourth grade I was annoying. I might still be annoying, but I’ve now surrounded myself with people who can tolerate me. In public school in rural Maine, I lacked this luxury. I kept raising my hand because I knew that when I got a question right the teacher was forced to give me validation. I was eager for validation from the authority figures in my life.

They saw that I was distracting other students so they gave me the textbook and a calculator and banished me to a back corner of the classroom. I was told to just figure out how to do all the questions at the end of each chapter. I did well in math that year, except that when my parents came in for parent-teacher conferences my teacher explained that I was letting the future I had in math slip through my fingers. My parents asked how, and my teacher elaborated: “Because Nisse is learning on his own; he’s refused to read the vocabulary sections of the textbook. He therefore has started calling a ‘sum’ an ‘add-em-up.’” Well, duh. “Sum” didn’t make sense because it wasn’t some of anything. It was all of it.

Nobody explained that sum is an abbreviation for summation which comes from the same base word as summary. Also nobody explained that the subtle dictatorial method of decision-making that drove the classroom forward was an inoculation to the militaristically inspired society in which I was meant to become a soldier.

Presidential campaign ads consistently refer to a “button” that can launch nuclear weapons, themselves always a kind of final solution. This concept of a touch-key that allows you to wipe clean all progress in other directions and focus simply on the destination is a relatively new concept that is intrinsic to the invention of the handheld calculator.

Our original western public education is a direct reaction to a compulsory military draft. The government saw the need to acclimate children to this style of taking orders and organizing themselves, so there were strict rules around discipline and punishment, and all of the moments of autonomy (food, play) were organized around lining up; they were allowances that could be removed with demerits. By World War I, the U.S. realized that wars won on the battlefield were won first in factories; there became a push to change the format of education to mimic factory work. Schools integrated bells to indicate when to change tasks, and classwork began to mimic the Ford model for repetitive tasks, and children, organized by age, were systematically awarded certificates of approval.

Post WWII, President Eisenhower warned the American people that “This conjunction of an immense military establishment and a large arms industry is new in the American experience. The total influence — economic, political, even spiritual — is felt in every city, every statehouse, every office of the federal government.” What he didn’t say was that the military was already adapting to a new world post atom bomb where wars were won with anticipatory fear: Awesome displays of massive destruction was the format of war in the cold war era. In response to the Soviet Union’s first atomic bomb test in 1955 and the launching of Sputnik in 1957, the Eisenhower administration passed the National Defense Education Act of 1958, which created higher standards for math and science as well as a student loan system specifically for students going to college for science, mathematics, engineering, or a modern foreign language — disciplines that were thought helpful in the fight against the red menace. The bill dedicated $70 million to new equipment in schools that helped teach science, math, or foreign language, and even held institutions to an oath, to “solemnly swear (or affirm) that [they] will bear true faith and allegiance to the United States of America and will support and defend the Constitution and the laws of the United States against all its enemies, foreign and domestic.” Y’know, in case you didn’t get what the money was for. In the ’60s, education officials, in their attempt to compete with the Soviets, changed the number of years of math required in high school from one to three. The “Rockefeller Report on Education” called The Pursuit of Excellence: Education and the Future of America (1958) is touted by educational and cold-war theorists as the paper from which the NDEA got its ideas for the bill that forever shaped public education in the U.S. Written by liberal intellectuals of the time, it reasons, regarding affirmative action, that “[In the case of African Americans,] it’s not so much that developed talent is rejected but that talent is not allowed to develop.” The authors also pointed to the unfairness of society’s demands for women to be wives and mothers without being given the opportunity to join the workforce. And yet, the same paper also claims that “The educational system, among other things is a great sorting out process,” and “The general academic capacity of students should be at least tentatively identified by the eighth grade as a result of repeated testings,” before ultimately externalizing the neo-liberal’s true agenda: “The heart of the matter is that we are moving with headlong speed into a new phase in man’s struggle to control his environment.”

When the first pocket calculator came out, the equals sign was combined on the same button with the addition sign and was philosophically linked to the answer — a relieving finality

By the mid ’70s, the Red Scare had died out, but as authorities were eager to maintain a delusionary commitment to the triumph of “good” over “evil,” the military-educational complex was given the gift of the handheld calculator — a machine made of buttons that could train students in the art of final answers, that only displayed conclusions, and that was accurate enough for scientific purposes but not invested in the truths that math had historically been based on. While Pythagoras had once drowned a man for suggesting that there might be numbers that exist that cannot be represented as a ratio of the two legs of a right triangle, now we were comfortable with formatting all numbers in decimal form and rounding to nine digits or less. Instead of 2/11, teachers became comfortable with “about 0.22.” I’m not advocating for drowning your enemies, but maybe a little passion for the concept of truth. As Georges Bataille writes in Inner Experience, “no answer ever preceded the question: and what does the question without anguish, without torment mean?” It becomes the job of only “good apostles,” not of philosophy, to “have an answer for everything; they indicate limits, discreetly, the steps to follow, as does, at burial, the master of ceremonies.” Can we not handle an argument without a final answer?

In 1971 when the first pocket calculator came out, the equals sign was combined on the same button with the addition sign and was philosophically linked to the answer. By 1974, calculators began losing the equals button altogether and replacing it with the more direct “enter” key. Now, on the graphing calculators most high schoolers use, the equals sign is either labeled “ANS,” or “EXE,” linking the execute button with a relieving finality.

When I asked my high-school students about the pros and cons of the calculator they eagerly told me that while it makes them “lazy and dependent” it is an “efficient tool for solving as many problems as possible as quickly as possible.” What problems? How many miles Sally has to run in order to catch up with Robert who has been running for two hours before Sally but runs at a slower pace? Why is it important to solve that quickly? My students offered the rebuttal that getting more practice is necessary if they are going to do well on the SATs. This gets to the heart of the cyclical nature of our standardized testing: Standards change the way we educate, which changes the way we create standards — standards beget standards.

Nowhere is this more evident than in the change in math education after the invention of the calculator. Never was a technology more quickly integrated into the curriculum. The calculators were heralded by the Mathematics Teacher as “pocket-sized units [that] are already replacing textbooks in elementary schools,” who went on to say that “teachers are hoping that what once seemed to children a tedious labor may, through the calculator, become fun.” And yet instead of being content with the students solving the same number of problems at a quicker rate, education standards adapted and changed what was expected of students. In what seems to be a parody of the way capitalism adapted to Keynes’s prediction that human work would be unneeded in a future of robots, we added more questions to every worksheet — we made the numbers in the word problems more confusing. We turned to more “real life examples” of balls falling off cliffs and gardens doubling in area. The real reason these examples popped up is, obviously, not because more balls were falling off cliffs nor were more landscapers doubling the area of the gardens they were building, but because squaring and square-rooting became simple with the invention of the calculator, and because there was a new tool that sped up old processes, educators excitedly demanded that students become proficient in the use of that tool. More important to the military-educational complex, all of these problems involved hitting the “execute” button in order to find the solution.

The corruption of the equals sign was the most complete change of identity of a symbol until the hashtag, which was originally a British symbol for the pound (weight) called the Libra, from the balancing scales of astrology. Robert Recorde — a Welsh mathematician from the 1500s with the same deep, sullen eyes that seem to hide a sadness he doesn’t think worth talking about as Milton from Office Space — dedicated all of the books he wrote to the queen, his chief aspiration being to take on the position of lead accountant at the treasury. He was obsessed with Arabic mathematics and was bent on bringing Arabic numerals to England. In his crusade to systematize data, he invented a symbol meant to describe the idea that two mathematical descriptions are equivalent: a set of two parallel lines, the “equals sign,” would signify the scales being balanced — that theirs are two equivalent sentences. He wrote that he wanted to use “a paire of paralleles, or gemowe lines of one lengthe: =, bicause noe 2 thynges can be moare equalle.” “Gemowe lines,” like the hashtag’s Libra, comes from an old astrological text that Recorde was reading, referring to an old symbol for the Gemini twins through two parallel lines.

And yet a recent study shows 6 to 12-year-old students see the equals sign as the necessary precursor to the answer in a mathematical sentence. Students will read 8=5+3 as being “written backwards.” The equals sign has taken on a brand new identity as the “solution sign.” The shifting of meaning of the equals sign is a movement to take away the mathematical teaching of the idea of equality and replace it with the mathematical teaching of the idea of finality. Instead of giving students the metaphor that on either side of equality is a set of contexts and thoughts that combine in different ways but mean the same thing, we give them the metaphor that if you do things in order you end up getting an answer, and then you are done.

Guy Debord was fascinated by idea that there could be a journey with no destination and created a practice of wandering through the city he lived in that he called the dérive, which is a term taken from sailing to describe the act of letting down your sails and drifting with the water.

Freshman year of college, my TI-83’s batteries died. I knew that you weren’t supposed to throw batteries away in the regular trash but I didn’t know where the battery trash was. At 18 years old I was very into the idea that I was a self-sufficient adult who needed the help of nobody but his own brain and so I didn’t ask what to do with batteries. Instead I just left them in my calculator. That was 12 years ago. I still have that calculator. The batteries still don’t work. I still don’t know what to do with old batteries. I still harbor the delusion that I am a self-sufficient adult who needs the help of nobody but my own brain.

I majored in math, which meant that for multiple tests I would do six-digit long division in the margins, and I would force myself to find ways to simplify algebraic expressions so that they were manageable fractions. I look back at my probability class as the one in which I learned the most useful tools for teaching number sense to high-school freshmen. I learned more through the process of decoding large numbers and breaking them down into their parts than I had in all of high school. The process taught me this, not the product. And yet I would not have engaged in this process without the product as a destination to lead me there.

Numbers were an invention, like the calculator, and we used them to investigate big questions. But they became less tools to manipulate tools, and more tools to manipulate us into thinking we had answers

In the 19th century, math education was a mental exercise. It was the mental equivalent of gym class, with the idea that the brain is simply a muscle needing to be stretched. In 1893 an educator said of arithmetic: “It is refreshing to know that there is one subject which [the student] must master for himself slowly, sometimes painfully, and always with much labor.” And A. Lawrence Lowell, the president of Harvard, in pointing to the purpose of math education, declared: “To those of us who have not pursued the study of mathematics since college days, the substance of what is taught to us has faded away, but the methods of thought, the attitude of mind and mode of approach have remained precious possessions.” In the early 20th century the debate of what is math education began orbiting around a divide between “pure” and “applied” mathematics. There became two schools of thought: G.H. Hardy’s perspective that Pure Mathematics is an art because “a mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” Applied Mathematics, to Hardy, was an exercise in hypocritical futility because “It is indeed rather astonishing how little practical value scientific knowledge has for ordinary men, how dull and commonplace such of it as has value is, and how its value seems almost to vary inversely to its reputed utility.” On the other hand, E.H. Moore, who presided over the American Mathematical Society, decried what he saw as a “chasm” between pure and applied mathematics. “Beginning with matters ‘thoroughly concrete in character’ they would proceed to … demonstrations of physical phenomena ‘to develop on the part of every student a true spirit of research, and an appreciation, practical as well as theoretical, of the fundamental methods of science.’”

Where Hardy wanted to allow math to be an art form devoid of applicability, since “a science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life,” Moore saw science as an avenue into math. He believed that the only way to get students interested in the abstractions was through an investigation of the practical — that a journey needs a destination to motivate the journey.

When I was in college I tried to nickname myself “NisAdventure” because my name is Nisse and I loved the idea of getting lost — of finding the joy in the process without the product. I didn’t see the irony of doing so by attempting to nickname myself. You can’t declare a nickname for yourself because a conclusion can’t be made without the process to get you there — the destination doesn’t come before the journey. Also you can’t declare a nickname for yourself because trying to do so makes you a narcissistic asshole.

Moore ended up winning out the debate over Hardy in determining the direction of math education in the U.S., and we have since seen math as a necessary class for learning how to balance your checkbook and creating engineers. And because the goal of math class has become usefulness, teachers search for ways to make math class more useful, which reinforces that the purpose of math class is to be useful. I find myself at odds with curriculum and unable to engage in the discussions that got me interested in math in the first place, because the students and other teachers are frustrated that my class is “not getting enough done.” There is a focus on results, and this stems from and to every part of education and society. We have a system that is focused on the proof of truth through numbers partially because we believe that numbers give us a perspective on truth. And yet any mathematician will tell you that the numbers are tools, not answers. Numbers were an invention, just like the calculator, and we used them to investigate the big questions, but at some point we were given these calculators to the end of spitting out answers. And what they spit out were numbers written in base 10 fractions that were rounded to nine fractions. These became less tools to manipulate tools, and more tools to manipulate us into thinking we had found the answers we were looking for.

A life with answers is not a life worth living; a life with answers is over. If math can teach us anything it should be that any good question simply leads to more questions, and yet the calculator has given us the delusion of finality: it has forgone exploration for the sake efficiency; it has offered endings in the place where a beginning may have birthed; it has turned play into work and made Jack a dull boy; it has morphed our brains into machines that process ideas as being “right” or “wrong.” And while I often feel comfortable with my versions of right and wrong because I surround myself with people who I like and who like me, I know I’ve lost friends. I know I’ve alienated people. And I know those people exist in the same world as me, believing in a different set of rights and wrongs. And if we don’t find a way for our rights and wrongs to co-exist we will be forced to prove ourselves righter or wronger. And I wonder who will win. Probably whoever has the biggest gun. Probably not me.

Nisse Greenberg is a storyteller, math educator, and vegetarian (ancestrally). His playground of work can be seen at nissegreenberg.com.