When I decided to learn college algebra on my own at age 37, I bought a textbook on the internet for less than five dollars, a dense model from the 1990s by HarpersCollins—something not unlike the ones that made me cry as a teenager. Coldplay, Clay Aiken, and College Algebra could be the alliterative title of a tragic and insufferable autobiography centering my personal adolescent trauma.
The opening section of my budget conscious textbook functions as a review of the entire history of mathematics to the seventeenth century. Meant to be a catch-up, a refresher, these eleven pages occupy my thoughts while I do dishes, walk the dogs, and try to sleep.
After all, we learn about things like gravity and zero and space and time when we are small children, treating them as educational building blocks rather than vast, evolving journeys of discovery. We assume that the infancy of human thought is correlative to the physical age of modern humans, teaching children how to read a clock but never explaining the contextual weight of a second. These ideas are, after all, highly complex—ones that took hundreds and sometimes thousands of years and many contributors to arrive to.
Although the first chapter is titled “Algebraic Expressions,” my textbook opens with the first section on real numbers and various terminology that is foundational to understanding what numbers really are. That’s the trickiest part, the thing we move fastest through, the meaning behind the concepts. We count out numbers with pieces of macaroni. But, see, numbers aren’t just what we can hold. That’s been the problem since the beginning.
The term “algebra” is attributed to the ninth century Persian mathematician Muhammad ibn Musa al-Khwarizmi, the first to develop a system solving equations by moving terms from one side of the equation to the other. For example, adding five to both sides of the equation in {x-5=10} gives a reliable and predictable correlating answer: x=15. This is what algebra is—trying to make sense of what is happening on two different sides of a line—the balance of one thing against the other, simplifying complexity and merging dualities.
The trouble is, as you will soon see, when positive and negative numbers interact with one another—when free will and fate collide, when integers entangle across time and space, causing students and characters to scatter across a gym floor—complexity is far more interesting.
Most textbooks, in any discipline, operate like a number line. There are two primary reasons: 1.) The history of Western academics is both linear and separatist—each subject—math, science, art, home economics, physical education, etc.—in its compartmentalized box. Brazilian theater director and activist Augusto Boal does a brilliant job of exploring how Aristotle’s failure to examine his own cultural lens created a framework for the last 2400 years of disciplinary isolation and idolization of perfection. Political influence in education ensures the continuation of this model. Math is built upon a temple of holiness and wholeness. Pythagoras saw to that. Incorporating other ideas is to muddy already murky waters, or so it seems.
2.) A college math teacher needs to get you to understand the principles as quickly as possible. The why it feels superfluous to spend a long time talking about positive and negative numbers. Many education models are built upon the idea of scaffolding, starting with something and building upon it gradually. It’s a tried and true model for solving equations partnered with the “I do, We do, You do” structure of teaching. Scaffolding skills often feels more important than scaffolding thinking, especially when problem-solving is the primary way for students to exhibit learning on standardized tests.
But the reality of most learning is that it is not hierarchical at all. It is about shifting from simple knowing to connection to transfer and back again: trying things, getting frustrated, getting lost, finding a point of entry, trying something else, learning a new piece of information, getting frustrated once more, and discovering; on and on again in a circle. It’s endless cyclical convergence: strange loops, Nietzsche’s eternal recurrence, Jung’s individuation, Hegelian dialectics, and the line between knowing and not knowing: via negativa and via positiva. Every few years, thinkers fracture thought and make it whole again—not unlike the process of learning itself.
The fundamental problem with Aristotle’s model of linear thinking, one he inherited from cult leader Pythagoras and others, is that it fails to allow for what true learning is: complexity and making meaning through convergence. The reason Western thinkers keep tinkering with the foundation of their ideas and arguing about Forms, taking them apart and putting them back together again, is because they’re only ever playing with one part of the puzzle.
Come to class with me.
Your teacher (someone kind and confident and so smart it makes you tingly) has stretched a line of tape across the gym with a mark in the middle for zero. You like this class. In our mythical academy, you are not any specific age. We are going to move through thinking for any age, any content. Your classmates are funny and weird like you, diverse and timeless.
Maybe this gym doesn’t look like your high school gym. It looks like whatever you want it to look like. Maybe it’s a field. Maybe it is snowing like it is right now outside my window. You know you can leave whenever you want—go pee, take a nap—and come back. The lesson will be here, waiting for you.
All the students stand in a straight line facing the teacher along the zero point in the center of the tape, forming a large plus sign across the floor of the gym.
First, the teacher calls out numbers and has the class take steps according to a simple key: left=negative, right=positive.
The teacher chimes, “Negative four!” Each student takes four steps—big or small—to the left. Already you notice that four to you is not four to your neighbor. Each round, you return to center and practice again.
In the next game, students stay on the called number, adding and subtracting the next integer. The called numbers offer a simple equation: -5+7=2. When students take seven steps to the right, they calculate and write the answer on a white erase board. After a few rounds, the teacher is confident that all students understand the connection between their physical movements, the number line, and an equation involving positive and negative numbers.
This understanding is enough, but our lessons rarely have endings or assessments. Instead of stopping the lesson, the teacher asks a question: “How do negative and positive numbers interact with one another? How do they both contribute to the final answer?”
What if after the lesson, your teacher asks you to identify the themes of a story you read together recently: Romeo and Juliet. The teacher tells you that the theme needs to be expressed as a duality. After discussing the play for a minute as a class, everyone picks “fate vs. free will.” The teacher then asks the class to make a decision about which side of the duality is positive or negative. After some discussion, your group picks “free will” as the positive.
Then, each student develops an equation with ten numbers (all between 1 and 10), positive or negative according to plot of the play. Each point represents a moment in the plot, and each number represents movement toward one of the two: free will or fate. Students make their own timelines and decisions about values.
When Romeo decides to go to the party and encounters Juliet, you add five—you think he only goes because he was persuaded by his friends. That’s not fate. That’s just being a human. But your neighbor subtracted three. When you ask her about it, she says it was fate for Romeo to be at the party.
When everyone finishes writing their equations, the teacher asks the class to perform them in steps in the gym across the number line. While the students act out the numerical motions of the play, the teacher records the whole class and shows the video later. You discuss with your friends what the composition looks like with everyone acting out their individual plot versions. You look at the long length of the line and consider how some students landed on the negative (fate) and others on the positive (free will) side. You wonder whether how people interpret narratives impacts the ways they think about and interpret numbers.
Your teacher repeats the question: “How do negative and positive numbers interact with one another? How do they both contribute to the final answer?”
My, friends, there are still so many stories to tell. After all, I haven’t even addressed these illusive references to Pythagoras and his personal Jonestown. The cyclical nature of learning means each question leads to another. Subscribe if you’re wondering:
If both positive and negative numbers contribute to the final answer, then what does this say about the nature and meaning of dualities? Where do dualities emerge in other disciplines? What determines the positive or negative value of something? How do culture or perspective contribute to the ways in which we estimate value? What does our exercise say about the nature of fracturing? What is in the space between integers and plot points or in the vast field beyond and around the line? What is the line itself?