The Most Useful Algebra Lesson Of All

I first encountered algebra in the sixth grade. Numbers disappeared–or at least, were consigned to secondary importance–and letters, mysterious ones like x, y, z, took center stage.  A mathematical expression called the ‘equation’–an incomprehensible sentence underwritten by an esoteric grammar–emerged on my intellectual horizon. (Strictly speaking, my teachers were rigorous enough to call these things ‘linear equations’ but all I could remember was the second word.) The rules for manipulating it, and triumphantly emerging from these machinations with the value of x held high as a trophy for all to see were, as my descriptions above might indicate, utterly incomprehensible to me. I stepped into the water and I immediately floundered, casting about in panic.

Unwilling to seek help outside the confines of my classroom–I did not press my brother, two grades senior to me, or my parents, for assistance with my lessons or my homework, which I was often submitting incomplete and incorrect–I was setting myself up for disaster. The axe fell eventually. In the first of the year’s so-called ‘terminal’ exams–because they were staged at the end of an academic term–I obtained a grand total of sixteen ‘marks’ out of hundred. It was an ‘epic fail,’ long before that term had acquired any currency.

Unfortunately, my fame in this domain of academic achievement did not, and indeed, could not, go unnoticed. My grades were noted on a ‘progress report’ and I was asked to bring it back to school, duly signed by my parents.  When I, hoping to escape the wrath of my father, showed it to my mother, she took one look at my math grade and told me he would sign the report instead. (This transference of responsibilities reflected a traditional division of parental labor when it came to my education; my mother helped me with the ‘humanities,’ my father with the ‘sciences.’)

My ‘interview’ with my father did not go well. He was perplexed by my grade, but even more so by my exam answer-book. I had executed some bizarre, inexplicable mathematical maneuvers, strewing symbols and numbers gaily all over its pages, thus allowing my teacher to grant me a few charity points for visible effort. Most embarrassingly, my father was able to surmise I had cheated, for I could not explain why certain moves had been made by me. (Indeed, I had; I had sent several panicked sideways glances at my neighbor’s answer-book during that fateful exam.) My mother sat close by, watching this interrogation–and my discomfiture–quietly. I could see my father’s visage tautening, his nostrils flaring. A stinging slap that would inflame my cheeks and set my ears ringing was probably headed my way. This was a man who had brooked no incompetence in his subordinates in his days in the air force; he would not stand for this display of stupidity and confusion on my part.

My father finally spoke, “Go get your maths book.” I complied. My father pulled out a notepad and a pen, looked at me, and spoke again, “Algebra is easy if you follow the rules.” I had no idea what those were.

I soon found out. My father explained to me what variables, constants, and coefficients–fractional and whole–were;  he told me I had to “bring all the variables to one side, and all the constants to the other”; I was supposed to “change signs when you change sides”; and so on.  It was not smooth sailing: on one occasion, after I had failed, yet again, to internalize one of my father’s instructions and committed a howler, he, overcome by exasperation, turned to my mother and confessed he would like to throttle me. I quaked and quivered, but he did not make good on that threat. He did though, tell me he would not let me go to sleep till I had mastered the art of solving linear equations.

The night wore on. My mother went to bed. My father and I continued to work through one problem after another. Slowly, algebra became comprehensible; indeed, it made perfect sense, and even began to appear as a little bit of a lark, a sleight of hand, a riddle with a key that could be made to work for you, and not just wizards and magicians. It was entirely plebeian; the masses could partake of its pleasures too.

Finally, my father assigned a set of problems for me to solve and bade me go into the living room to work on them by myself. If I solved them correctly, I could go to bed at last. I got to work; my father began his bedtime ritual of changing clothes and brushing his teeth. A few minutes later, he opened the door of the living room to check on my progress: Was I moving along? I said I was.

Once I thought I was done, I took my work over to my father. For a minute or two, he sat there, looking impassively at my scribbles. Then, he looked up and said, “Good work; go to sleep. You’ve got it.” I complied again.

I never became a mathematician. But I never feared the ‘lazy man’s arithmetic’¹ again.

Notes:

  1. Legend has it that this is how Einstein’s father explained the heart of algebra to him: ‘you just act as if you know what is.’

The Abiding ‘Mystery’ of Calculus

I first encountered calculus in the eleventh grade. A mysterious symbol had made an appearance in my physics text–in the section on dynamics–as we studied displacement, velocity and acceleration. What was this ds/dt thing anyway? I had, at that point in time, never studied calculus of any variety; to suddenly encounter a derivative was to be confronted with mystery of the highest kind. I asked for explanation and clarification; I received less than satisfactory obfuscation in response. Something about ‘instantaneous rate of change’, whatever that was.

A few months later, having encountered differential calculus in the mathematics syllabus, I was considerably, if not totally, edified. Functions, curves, graphs, tangents; somehow, I was able to partially relate the material we had studied in the physics class to this mathematical paraphernalia. And then, a little later, in the twelfth grade, having encountered integral calculus and then differential equations, other pieces of the puzzle fell into place as the relationship between mathematical apparatus, the models they comprised, and the physical world became a little clearer.

But as the story of my introduction to calculus–an abrupt exposure to its application and formalism in dynamical analysis–shows, calculus had an initial air of mystery that took some shaking. It had been suddenly introduced as a mathematical tool to enable grappling with a problem of physical mechanics, but the formal insights that lay at its core–especially the concept of a limit–were decidedly unfamiliar. More to the point, its use seemed utterly gratuitous; I could not see how my understanding of the physical details of velocity and acceleration had been improved in any way. And even when I did study differential calculus, I felt as if I became an expert manipulator of its many recipes and techniques well before I understood what my activity entailed. Syntactical manipulation, the transformation of one set of mathematical symbols into another according to a well-specified algorithmic procedure, was easy enough; understanding what those meant, and how they underwrote our understanding of the world of becoming and change, was a different matter.

We were science students in high school, ostensibly preparing ourselves for careers in engineering, medicine, and perhaps even basic research in the physical sciences; calculus was one of our most important tools. But we remained befuddled by its place in the conceptual apparatus of our studies for a very long time. This should be, and was then, a matter of some perplexity, especially when I consider how enlightened I felt when I better understood its place in making a changing world comprehensible.

Years on, when I became embroiled in debates over curricula in computer science undergraduate education, it occurred to me little had changed; many students remained perplexed by calculus’ importance in their education, by its most foundational presumptions and applications.Nothing quite exercises pedagogues like mathematics education, and in their catalog of perplexities, the failure to properly contextualize calculus should rank especially high. I’m almost tempted to describe it as a civilizational failure, so convinced am I of the judgment of any extraterrestrial visitors when confronted with this peculiar combination of indispensability and incomprehensibility in our epistemic scheme of things.

Manil Suri on the Beauty and Beguilement of Mathematics

Manil Suri has an interesting Op-Ed on math–How To Fall In Love with Math–in The New York Times today. As befitting someone who is both a mathematician and a novelist, there are passages of writing in it that are both elegant and mathematically sound. The examples he provides of mathematical beauty–the natural numbers, n-sided regular polygons that become circles as n approaches infinity, fractals–are commonly used, but for all that they have not lost any of their power to beguile and fascinate.

I fear though that Suri’s message–that math is beautiful, creative, elegant, not to be confused with routine number crunching, and worthy of wonder and exaltation and careful study–will not be heard through the haze of the very math anxiety he seeks to cure. Suri needn’t feel too bad about this though; many others have tried and failed in this very endeavor in the past.  If, as Suri suggests, we are wired for math, we also sometimes give the appearance of being chronically, congenitally, incurably anxious about it.

A personal note: I came to a realization of mathematics’ beauty late myself. Like many of the math-phobic that Suri refers to in his article, through my school years my attitude toward the study of math was fraught with fear and befuddlement. I was acceptably competent in the very junior grades but a harsh teacher in the seventh grade ensured that I would earn my worst grades then. My concerned father took it upon himself to drill me in algebra and I regained a little confidence. Not enough though, to want to study it at the higher levels that were made available to us in the ninth and tenth grades. But the pressure to study engineering at the university level meant I had to return to the study of more advanced mathematics for the eleventh and twelfth grades.

In those two years, I was exposed to calculus for the first time and came to love it; its connection with motion, the slopes of curves, the use of differential equations to model complex, dynamic systems; these all spoke to me of a logical system deeply implicated in the physical world around me. Once I had learned calculus, I saw it everywhere around me: in a stone’s dropping, a car’s acceleration, a rocket’s launch, an athlete’s push off the starting block.

I majored in mathematics and statistics at university, but if I saw any beauty in mathematics in those years, it came when I saw some good friends of mine working out complex problems with some style. I had lost motivation and interest and barely survived my college years. My graduate studies in computer science didn’t help; I was able to–unfortunately–skip the theory of computation and graduate.

Years later, when I encountered mathematical logic as part of my studies in philosophy, some of my appreciation for the world of abstract symbol manipulation came back. It helped that my dissertation advisor was an accomplished mathematician whose theorems and proofs sparkled with style and substance alike. From him, I regained an appreciation for the beauty of the symbolic world.

I was never a mathematician, and don’t work in logic–mathematical or philosophical–any more. But my brief forays into that world were enough to convince me of the truths that Suri refers to in his piece:

[Mathematics] is really about ideas above anything else. Ideas that inform our existence, that permeate our universe and beyond, that can surprise and enthrall.

Andrew Hacker on the Supposed Superfluousness of Algebra

An Op-Ed titled ‘Is Algebra necessary’ is bound to provoke reaction. So, here I am, reacting to Andrew Hacker’s anti-algebra screed (New York Times, July 29th, 2012). It is a strange argument, one unsure of what it is attacking–mandatory math education, elementary algebra, higher algebra?–and one founded on an extremely dubious premise: that the way to carry out educational reform is to cherry pick your way through a curriculum, questioning the ‘utility’ of a particular component in case there are no jobs that require an exact application of its material. Hacker makes things worse by leaning on statistics that cry out for alternative explanations and pedagogical reform, rather than the ‘lets drop the subject students seem to have difficulty with’ approach that he favors. If American students are struggling with algebra, it might be time to inquire into how it is taught, to show students how abstraction and symbolic representation are key to understanding a modern world underwritten by science and technology. Dropping algebra seems like a profoundly misguided overreaction.

The ‘surrender in the face of poor test scores’ approach results in a series of bizarre statements of which the following are merely representative samples:

It’s true that mathematics requires mental exertion. But there’s no evidence that being able to prove (x² + y²)² = (x² – y²)² + (2xy)² leads to more credible political opinions or social analysis.

Certification programs for veterinary technicians require algebra, although none of the graduates I’ve met have ever used it in diagnosing or treating their patients. Medical schools like Harvard and Johns Hopkins demand calculus of all their applicants, even if it doesn’t figure in the clinical curriculum, let alone in subsequent practice.

I have news for Hacker. There is little evidence that being able to leads to ‘more credible political opinions or social analysis’ either. Furthermore, if job skills  are examined as superficially as Hacker does in his examples then it becomes all too easy to dismiss large parts of one’s educational background as being irrelevant. Hacker would be alarmed, I presume, to find out that even though modern physicists hardly ever roll balls down inclined planes, freshman physicists are still required to spend a semester solving problems that are full of problems that stress just that. Hacker dismisses the argument for a general education in mathematics with the alarmingly glib ‘It’s true that mathematics requires mental exertion’ without stopping to inquire what that ‘mental exertion’ might consist of and what it might engender in turn. This is hardly an attitude toward pedagogical reform that breeds confidence.

It is a consequence of Hacker’s argument that the only students who should receive an education in algebra are those preparing for careers that require them to apply algebraic techniques and concepts in their jobs. Everyone else can be spared its ‘difficulties, ‘ like the above-mentioned abstraction and symbolic representation. How would an extension of this argument work in, say, fields like history or literature? A relentless whittling down of the curriculum would result, leaving us with a list of subjects read off the Help Wanted Ads section.

This is an impoverished, grimly utilitarian, and ultimately soulless view of education.