I’m about halfway through Eagleton’s book, Hope Without Optimism. Optimism, it seems, is a vacuous, ungrounded belief that everything will be ok. Everything will work out for the best. It’s an attitude and an outlook. And when things don’t work out for the best, they have to be quickly recast as actually the best but we didn’t know it right away (the “what doesn’t kill you makes you stronger” kind of thing). Eagleton is not a fan of optimism. Neither am I.
Hope, on the other hand, requires a lot of work and for the most part, has very little to do with optimism. You can be hopeful without being optimistic. You can hope desperately for something, knowing full well that it likely won’t come to pass. And hope requires that we acknowledge that things might not work out for the best. That doesn’t mean we should quit. Not at all. There is a realism and a drive implied with hope that optimism doesn’t require. Hopeful people hold a certain fidelity to the virtue of hope. The whole idea is to forge ahead in a hopeful way, despite the presence or lack of optimism.
I’ve begun MATH 140 for the second time. So far, the material is familiar and I am passing, but I’m certainly not “acing” this course the way I expected–given that his is my second time through. And, of course, life is again getting in the way in all kinds of wonderful and insidious ways.
However: I will remain hopeful. Not optimistic. But unflaggingly hopeful.
The summer math class ended in disaster, or what I (as a teacher) might have previously called “a teachable moment”: I dropped the class on the last day possible because I tried to build success on the foundation of a Bad Plan.
I ended up with this Bad Plan for Good Reasons. These Good Reasons may sound vaguely like excuses. But they’re not. Totally not. I’m detailing them here because they make me feel good.
After registering for Math Class, I was invited to teach one of my favorite classes during the summer session. How could I NOT teach this class?? I love this class. It grew into two sections. And three concurrent summer classes, whether teaching or taking, is too many.
Then, I was awarded a research grant to visit Ukraine during the fall. A colleague who speaks Ukrainian offered to travel there together, but she was going the week before the summer session. How could I NOT go with a native speaker who could help me translate?? So I flew home the day Math Class started and drove right to campus from the airport, jet-lagged and delirious.
The week after this summer circus began, I was invited to participate in an edited collection directly related to my research. This was great news. How could I NOT agree?? But it included several summer deadlines, lots of intense thinking, and a few wine-infused conversations.
On top of all these Good Reasons, I was facing the Known Obstacles:
Summer classes pack 15 weeks of content into 7 weeks of time.
The class was held from 6:00PM to 10:00PM. PM. The middle of the night.
Math takes me a long time.
Math just really takes me a long time.
Lots of time.
Looking at all this now, it’s obvious that I should have dropped the class right away. My math instructor could see it. He was very kind and after a five-quiz failure streak, he gently suggested that I could consider dropping so that my transcript wasn’t saddled with an F. He suggested that I come by his office to talk about my math goals, because maybe there was a better way for me to achieve whatever it is I’m hoping to achieve. He said I could continue to attend class even after dropping, so that when I did re-enroll into a long semester, I would be as prepared as possible.
Hearing the news that, despite your best efforts, you are very likely to fail is not easy. Delivering the news is worse. But in a world where a single course costs over $1,000 and an F can cause immeasurable problems in a competitive job market, it feels irresponsible not to have this conversation. I’ve discussed Bad Plans with several students over the years because I could see the writing on their walls much more clearly than I could see it on my own. As a chronic Bad Planner, I feel more than a little hypocritical offering others advice on this topic.
I do know, though, that failure for a Bad Planner is relative. Did I fail to meet the requirements of the course? You betcha. But look at all the other things I got done while I was fretting about failing math. Many of us do our best work when we’re looking at it peripherally. And that means the risk of failing whatever we’re looking at head-on.
The fall semester starts next week, and I’m again enrolled in MATH 140. Have I learned anything from this summer’s teachable moment? Debatable. I am again over-committed in a dozen other ways. And I am again underestimating the time commitment. Because technically, I’ve already taken this class once. And this time, it’s in the morning. And I have 15 whole weeks. So I’m pretty confident that this semester will be a success. It just might not be a success when it comes to math.
After a two month hiatus from learning anything new, the summer session careened into view. MATH 140: Introductory Mathematical Analysis started over two weeks ago, and only just now, today, do I have time to take a breath. Initial observation: summer class timelines are not to be taken lightly.
This is the course in which we finish up the algebra begun in MATH 120, and we start trigonometry. How did I not know that trigonometry was just Super Fantastic Geometry? I vaguely remember enjoying geometry in high school, although my only concrete memory is the gift of a pencil that said, “A logarithm is an exponent.”
In trigonometry, we’re learning how to calculate the speed of airplanes, the velocity of tsunami waves, and the ascent of hot air balloons. We can also calculate a ship’s bearings and a satellite’s orbit. Of course, algebra is simmering under the surface of it all: even if I get the concept and pick the right formula, I can’t complete the calculation without stupid PEDMAS poking me in the eye again.* But the ability to look at a triangle on the page and see movement and change in the physical world is powerful. I am in love.
The ability to see movement and change through language is powerful too, and it’s one reason I was an English major from the get-go. These days, I write a lot of things without knowing my words’ trajectories once launched. This blog is one of those things. Many of my social media posts are these things, too. Someone blocked me last week because of a comment I made related to #BlackLivesMatter and the Dallas sniper. Now that I’m blocked, I can’t access the comment to delete or amend it, I can’t respond, and I can’t see what anyone else has written. That ship has sailed, and in which direction I have no idea.
Who knows what other trajectories my writing has taken or what kind of change I have effected, I hope more often for better than for worse. But in just two short weeks of trigonometry, this power pops all over again, clearly and consistently. I am excited to have been reminded that it is sometimes possible to see a large chunk of the world on a relatively small piece of paper.
*Look! I just tried to express a relationship between algebra and trigonometry, which I couldn’t have done six months ago. Related: I still don’t know what calculus means.
Mercifully, the semester has ended. People keep asking me (because I keep bringing it up) why I took a math class. Originally, I wanted to better understand some of my favorite theorists and writers. While that’s still true, I don’t know now if it’s entirely true. I’m not at all sure why I’m doing this. With all the time I spent on this course, wouldn’t I have been smarter to do something overtly career-related? Maybe. Probably. I don’t know.
In January, I planned to work ahead so I could linger over important concepts and make astounding connections. But that never happened. It was all I could do to keep up with the basics. I had too much else going on. Everyone does. A writing student wrote something similar in his reflection on my English course. In fact, he described it as “crushing.” I empathize, but I also disagree: a particular course is not crushing. It’s just one variable within a larger societal structure designed to present “crushed” as our natural state of being. (In other words: If my course hadn’t crushed him, something else would have.) Same goes for math. And even when a course is pared down to make room for lingering, students (including myself) will likely absorb that extra time into other, more tangible and measurable commitments.
Despite the difficulty in assessing a good linger, I nonetheless believe in its value. A thoughtful reflection can far outweigh the more easily quantified skills. And so here’s mine:
From the perspective of teaching and student-ing:Doing math problems together in class is super helpful. Sitting on the back row is, generally, the bad idea I always knew it was. Offering to help a student during office hours has huge impact, even if the student never actually comes to office hours. Test anxiety is real, and “eating a good breakfast” doesn’t help. Grading math tests seems to be as labor-intensive as grading essays.
From the perspective of learning: THIS WAS SO HARD. It was a lot of trial-and-error, repetition, and memorization. I’m not advanced enough yet to understand the whys, the causes, or the “meaning” in most of what we learned, and that made it even harder to commit a formula or process to memory. Note to self: You felt the same way when you were learning to knit and could only make square things. Eventually, you did knit a sock. Be patient.
From the perspective of math: With only the very tippiest tip of the iceberg under my belt, I see now that basic math is not the tight narrative I was expecting. I knew the advanced stuff would be hairy and imaginary and unpredictable, but I was naively expecting to find a solid foundation in this basic algebra class–I guess because the last time I tried to learn algebra was in high school where ideas are often presented as immutable Truths. Instead, I see math has the same bunch of tiny little truths with which postmodernism has littered the humanities. I should have known: it’s always turtles all the way down. Not to be overly dramatic, but this is causing some existential angst to flare up. Note to self: Take a breath. The world isn’t any less stable than it was this time last year.
What’s next: I have passed MATH 120a: Algebraic Methods somewhere between the skin of my teeth and the hair on my chinny-chin-chin. In the upcoming summer session, I’m taking MATH 140: Introductory Mathematical Analysis. Despite the awesome course title, I think it’s really just Algebra II since we’re using the same book as 120a. My (likely faulty) expectation is that 140 won’t be as difficult: I won’t have to do so much legwork to get caught up, and the math classroom won’t feel so unfamiliar. However, it’s 15 weeks of material done in 7 weeks. So we’ll see. I’ll check in here during the first week of July.
In the 1970s, I drew buttons and flashing lights on a cardboard box, cut slots on each end, and called it a computer. You could write a question on a card, drop it into the entry slot, and it would come out the other end with the answer to your query. The catch (a doozy): I had to write the answer on the card before you dropped it into the computer, since nothing actually happened on the inside of the box.
In math class, we’ve reached the chapter on functions. Functions do the kinds of things I very much wanted to make happen inside my cardboard computer. They are all about inputs and outputs: f(x) = [something crazy like 3x + 2]. f(x) — which is a fancy way to say y — is dependent on the input of x. If you input something as “x” and only one thing is output at the other end, then it’s a function and you can happily plot x and f(x) on a graph.
The linear model that we learned about last fall in statistics is closely related to all of this (or so it seems to me). The linear model includes a slope (of a line), a y-intercept, and a relationship between dependent and independent variables. They all work together to help predict the locations of dots on a regression analysis graph. And I do love graphs.
Understanding dependent and independent variables about killed me last semester, and they’re doing a number on me again. But the cause/effect is clearer this time around. My hat’s off to everyone who has to teach stats to someone like me, someone who doesn’t know what an algebraic function is. I think this must have been the same kind of frustration as playing Mad Libs with someone who doesn’t know a noun from a verb from a postmodern platitude.
I feel triumphant in making this connection between functions and linear models (even if I still have some of it wrong). And I am excited to think about how the spatial mapping of data (as dots) and relationships (as lines and slopes) is a much stronger undercurrent than I realized. Maybe I should have known it. But I didn’t. But now I do. So that’s progress, with the slope of my own linear progression again pointed upward and onward.
First: I did not fail the test. Nope. Not a failure. Not today. Voice of reason: Although I am happy to have passed, it was a genuine surprise. So clearly I have no idea how to self-assess my abilities in this course. File that away as something to think about post-celebration.
Second: I made progress on my black-hole tendencies. I asked two questions in class, and I bought a new book. Yes, this book. Yes, the cover pumps a personality quiz and “boy-crazy confessionals.” But Winnie-from-Wonder-Years’s tone is so much more likeable than my $400 course textbook. The blurb promises that she “shows you how to ace algebra and soar to the top of your class–in style!” The pep-talk to us girls (tween or otherwise) about being able to do math is a bonus.
Third: We started graphing things this week. MY OPTIMISM IS RENEWED. I love graphing. I love graph paper. I love charts and tables. And grids. It’s the whole reason I became a girl-scout as a kid: selling cookies meant I had control over the most magnificent, color-coded spreadsheet that a 1980’s 10-year old could hope for. And when we planted a garden, we went the square-foot gardening route because it required a grid. And crossword puzzles: a favorite pass-time. So many tiny little boxes. I use graph paper all the time, but using graph paper for its intended purpose brings a special kind of joy.
Finally: I’m not funny. I know it. Last week, a stranger commented that this blog was interesting but not funny. Being interesting can be hard, so I’ll take that as a win and continue to forge ahead. Onward and upward, everyone, mechanical pencils at the ready. It’s a whole new week.
We had our first test this week. The last question was a word problem about the amount of pollution in the air based on the speed of the wind. Somehow, I ended up with a negative wind speed. Which I know is illogical and impossible (unless I completely misunderstood the LIGO announcement this week). But the math that I had worked out to arrive at the negative wind speed looked pretty much right to me. And so, knowing that my answer was absolutely wrong, I wrote it down anyway. Something is always better than nothing.
It is only this week dawning on me that I might actually fail this course, which has put me into a very fast five-stages-of-grief tailspin. At the peak of my frustration, I told a friend that I was doing everything I knew how to do, and it wasn’t enough. She said:
“Did you visit your professor during office hours?” No.
“Did you seek out tutoring?” No.
“Did you find other books to explain the same problem differently?” No.
She pointed out that I was not, in fact, doing everything I knew how to do. Fellow math students have also suggested that I sit in the tutoring center while doing homework and that I ask around for links to YouTube videos of people working through the steps for given problems.
Honestly? I HATE THESE IDEAS. I believed learning math wouldn’t require as much collaboration as learning to write seems to require. And honestly, the idea of doing math in a vacuum appeals to me. I’m not sure why, when I know how powerful collaboration can be. For some reason, I want to be inside a black hole with math, where nothing IS something. Or something like that. Maybe my answer to this test question was me trying to tell myself as much. Except I don’t know enough math to have ever orchestrated the negative wind speed answer on purpose.
This week, I need to reassess my goals with this math business. If this project is going to last longer than a semester, I suppose I need a better plan. Working in a black hole, as much as I like the idea of it, isn’t going to get me very far.
A month into the semester, and my algebra book has not yet mentioned this critical bit: the two solutions produced by a quadratic equation are actually the points on a graph that a parabola passes through. Not until ch 3 this week, “Functions and Graphs,” when finally: we have some pictures. This changes everything.
Coincidentally, this week my own students and I read the part of Foucault’s The Order of Things where he mentions “the beautiful calligrams dreamed of by Linnaeus” (135). A calligram is a piece of text written in the shape of the object it describes. It’s often associated with poetry, but it’s also tied by definition to pictures.
Botanist Carl Linnaeus attempted to use calligrams in his scientific descriptions of plants: “the order of the description, its division into paragraphs, and even its typographical modules, should reproduce the form of the plant itself. That the printed text, in its variables of form, arrangement, and quantity, should have a vegetable structure” (135). Linnaeus felt that his classification system would be better represented if he used the lines on the page as both text and image. The idea of overlaying a mathematical, formulaic grid onto language in order to suss out buried meanings and connections is nothing new. Centuries later Lacan would try something similar (in my mind, anyway) by creating mathemes: graphic representations of his ideas that you can now buy on tee-shirts.
In a separate essay called “This is not a pipe,” Foucault discusses Magritte’s paradoxical painting as another type of calligram “secretly formed, and then carefully undone.” He writes that calligrams “bring text and image as close as possible to each other,” and usually the calligram erases the binary between: “to show and to name; to figure and to speak; to reproduce and to articulate; to intimate and to signify; to look at and to read.” In Magritte’s work, says Foucault, through the contradiction and the conflation of the words and image, this is an act of mischief.
The graph of a quadratic equation seems to be a mischievous variation on the calligram, one that conflates the idea of general and specific, of a formula to be applied universally and of a specific diagram of a particular banana. Seeing the equation and its result together simultaneously forms and undoes their relationship, at least for the uninitiated (as I am), at which point we are (I am) surprised and delighted to find the correspondence.
And a parting question for those who are already fluent in quadratics (can you say it that way?). I imagine that having both the equation and the graph is a bit redundant, the way Neo sees the Matrix code and the agents simultaneously, so once fluent, does the act of plotting the graph continue to generate any meaning, laughter, or surprise?
We have begun working with quadratic equations. I am failing this miserably, but I think I have discovered the root of the problem: the Order of Operations. This is the fundamental set of rules that dictates 5 + 6 x 2 equals 17 and not 22. I realized last week that when overwhelmed by an equation, I start with the parts that look easy (so I do all the addition first, for example). The Order of Operations says no, no, no, you can’t do it that way, and it makes my answers non-negotiably wrong.
Until today, I believed that basic math skills were grounded in axioms (self-evident propositions) and theorems (things capable of being proved). But in studying up on the Order of Operations, I found that it is a convention. Granted, it’s a widely accepted, super duper important one. But at its heart, a convention is simply “an agreement or covenant between parties.” In other words, I’m failing math because of a spit and a handshake.
If a convention is simply an ongoing agreement to a set of (arbitrary) rules, then the Order of Operations is just math’s grammar, a study of “the relations of words in the sentence, and with the rules for employing these in accordance with established usage.” My field of writing and composition has largely abandoned the right/wrong binary of grammar in favor of multiple grammars representing marginalized voices. My biased impression is that basic math has, conversely, tightened its grip on this binary.*
Math and writing used to have multiplicity more in common. In Hidden Harmonies: The Lives and Times of the Pythagorean Theorem, I read that Babylonian scribes may have “thought of values differently arrived at the way we think of variant spellings: there is no ideal that ‘color’ or ‘colour’ better realizes” (36). And the authors then write that the algebraic Babylonian culture “was founded on recipes, which are always modified by locality and detail” (41).
I have to tell you, I love the idea that my test answer isn’t wrong–it is simply my shiraz to math’s syrah. Come on, Aunt Sally. Let’s carve out a little space for my particular dialect of quadratic equation. We can discuss further over a glass of wine.
*This might be because so much of the homework is done online with inflexible, automated grading systems. The computer is a harsh mistress.
Chapter one of College Algebra: we are learning about square roots. After you square something, you can use a square root to undo the square. They provide reversibility. The number under the root-sign-thingy is the radicand, and the number on the outside is the index. The entire square root expression is called a radical.
Radicals seem to be mostly used for party tricks with negative numbers. “Hey, Pythagoras, watch me square this negative number. POOF. I have a positive number. Where did the negatives go??” Without the negative, you can do practical things like calculate a standard deviation and play with “degrees of freedom.” As long as the squared numbers remain under the safe cover of the radical, they are in a (heterotopic) space where you can add, divide, apply, and make meaning. And the really wondrous part? Not only can you square away negative signs when you don’t want them, and not only can you later conjure them back with your radical-at-the-ready, but you can ALSO bring them back with the new meanings still attached.
A radical in math is related etymologically to a radical in politics. The word means “from the roots, fundamental,” and—most surprising to me—”vital,” as in: “the humour or moisture once thought to be present in all living organisms as a necessary condition of their vitality.” Thus a radical math expression and radical politics both desire to get at the root of a thing, to dig down to the fundamental part that cannot be broken down into anything smaller, to understand the vitality permeating the matter. Imagining radical activists (think Arab Spring or Occupy) camping under the umbrella of the radical sign, it’s only a small step then to disappearing the negatives, functioning from a place of absolute value, and harnessing the inherent vitality of their expression. The power of this expression comes from the quantity of the root, which increases exponentially.
I’m not sure of the value in this comparison, other than to say there it is. In my studies of radical politics, I plan to think more about the disappearing negatives, degrees of freedom, and how meaning is carried back across the threshold of the radical.