What is Algebra? And is it memorable and useful enough to justify doing lasting harm to kids' lives over it?
This is a long-overdue response to Paul Cottle, a Physics professor at FSU, with whom I exchanged some very enjoyable thoughts and questions that culminated in this essay he wrote back in January. I think this is part of a vital discussion about the systemic purpose, meaning, and consequences of education. I hope we'll keep it going.
I have a terrible mechanical/computational memory. My wife and I often played "Memory" with our kids as they grew up. I never won. Not once. Ever. The locations of the little cows always swam on me once we turned them back over. My wife and kids got a giant kick out of laughing at my frustration.
Moreover, I can't remember phone numbers; names; where we keep the pots; when I've locked a door (multiple times a week, I make multiple trips to our doors at night because I can't be sure that I'm not remembering locking them last night).
Family ridicule aside, I have come to regard this poor computational memory as something of a gift, rather than a curse.
Because, on the flip side, I have a ridiculously good impressionistic/emotional memory. When something makes an impression, it never goes away. I internalize it -- and potentially weaponize it -- automatically. Without much effort, really.
For instance, names don't make an impression on me. But if that human whose name I can't remember communicates their pain or fear or love or anger or experience to me, it attaches itself to an understanding of that person whose name I can't remember.
I am not a brain scientist. I'm theorizing here only from my own experience. But I think my brain automatically sheds clutter it does not think it needs so it can lock in on the deeper thoughts and human concepts that make me who I am. I think my brain has an ability to sense and assign value in what it processes -- and discard what I don't really value.
It's hard for me to know -- or even speculate -- that people who remember more of what they encounter than I do have greater difficulty discerning importance and value. So I'll refrain from doing it.
But I'm increasingly thankful, in my life, for my computational memory shortcomings. I think they are a feature, not a bug, of any talent I possess and any accomplishment I can claim.
Memory, retention, and achievement
This question of memory has powerful implications for the way we evaluate education achievement. It's not abstract. I think it bears directly on one of the common complaints one often hears from colleges about incoming students: their inability to pass college Algebra opt-out exams and their need for remedial math.
For instance, the state believes that my son is quite good at Algebra. He passed the EoC in 8th grade with a score right on the line between very good and superb. As I understand it, he basically knocked out the toughest requirement for high graduation before ever entering high school. (He had a wonderful Algebra teacher, Yesenia Cruz, at Crystal Lake Middle. And she continues to offer her help to him and his friends today as a high school freshmen, when there is absolutely nothing in it for her to do that other than a general commitment to kids. She is fantastic.)
However, I have absolutely no faith, whatsoever, that my son is going to retain the Algebra skills Ms. Cruz helped him learn in 8th grade, nor those he will learn in 10th grade Algebra II, by the time he enters college.
I have no faith, despite what the state standardized test says, that he is going to pass any college Algebra placement exam. I think he is going to forget most of what he learned. I certainly did at his age.
And in this age -- the era of Common core, of sprawling curriculum maps, and high stakes testing -- he must remember a significantly greater volume of discrete math chunks than I needed to. The sheer volume of these "standards" makes them impossible to practice in an ongoing way over time. Indeed, the sheer volume made them very difficult to practice in an ongoing way -- as he was learning them. Virtually every day brought a new standard in a pellmell race toward an illusion of competency, one that our state happily perpetuates.
If you aren't retaining, you aren't learning...?
I made this point about volume, timing, and memory to a friend and college math professor recently. She said this in response: "I would argue that if you aren’t retaining the knowledge then you aren’t learning."
I have two responses to that.
What's the difference -- in that sentence -- between the meaning of "retaining" and the meaning of "remembering"?
I wrote a book about Florida between 1915 and 1930 called Age of Barbarity. It incorporated a million discrete facts I learned into a narrative I created about a massively complex story that no one had ever told before. The dance of variables that created this story might even be called a kind of Algebra. Yet, if I gave the book to my friend and told her to come up with 40-question test about dates, sources, highly detailed character actions, I doubt I could pass it. It took me three years to learn, process, and write this book. The volume of factoids to retain is enormous. And my memory couldn't wait to shed them. So did I not learn the narrative of my own book? Come talk to me about it and see. Again, this isn't an abstract thought experiment. I published my book in 2013, almost exactly five years ago. My son passed the Algebra 1 Eoc with a high score in May 2017. He will likely enter college in the Fall of 2021 -- four-and-a-half years later. Is he any more likely to retain the full enormous volume of Algebra factoids than I am likely to retain the factoids of my 500-page book?
We don't have to look very far to actually answer that question beyond Billy Townsend's experience. Of the many important points Ted Dintersmith has made about education, this one sticks out perhaps most powerfully to me. Follow the link here.
In my travels, I visited the Lawrenceville School, rated as one of the very best high schools in the United States. To its credit, Lawrenceville conducted a fascinating experiment a decade ago. After summer vacation, returning students retook the final exams they had completed in June for their science courses. Actually, they retook simplified versions of these exams, after faculty removed low-level “forgettable” questions The results were stunning. The average grade in June was a B+ (87 percent). When the simplified test was taken in September, the average grade plummeted to an F (58 percent). Not one student retained mastery of all key concepts they appear to have learned in June. The obvious question: if what was “learned” vanishes so quickly, was anything learned in the first place?
Forget five years. In just three months, the tyranny of memory attacked the illusion of achievement -- at a super elite prep school. It's the same exact problem my math college professor friend sees at less elite levels. It's a model problem; it's a conceptual problem; it's a how-we-evaluate problem. It's not a kid or a teacher problem.
And yet we have addressed it, as a country and state, by forcing kids to cram ever harder and measuring ever more punitively. We have addressed this not by modernizing our political conception of achieving; but by suggesting there's something morally and professionally wrong with the children and teachers trying to execute this madness.
And we do all this at a time when we can't stop talking about the AI and data mining capabilities of our emerging robot overlords. Whatever one can say about our robot overlords, they will not forget and they will not sleep. They far out-datamine us today; and the gap will only grow. Humans are not data miners. We differentiate ourselves in the open spaces between data -- in the negative data spaces where judgment matters. Yet, our STEM-based education system, especially in Florida, is set up as if success in life is like a game of Jeopardy.
Indeed, if Watson read my book in the nanosecond it would take it to do so, it would trounce me in a Jeopardy game about my book. If the state of Florida assessed me on the minute knowledge of my own book, it would conclude I didn't work hard enough on it. And it would defund me and call me a slacker -- and blame "union bosses" for it.
We don't call it Math-ing, Science-ing, or History-ing
We only make one core academic subject into a gerund, a noun derived from a verb: Reading.
Have you ever thought about why? Because literacy is different and more fundamental than anything else we teach. Literacy experts tell me that no one who learns to read in a native language loses the ability to read without a brain injury. Reading is the educational equivalent of riding a bike.
Math and Science and History are nouns, things made up of infinite, dynamic, intertwined acts of doing. Yet, we freeze all that doing into static replicas -- and then assess our kids on how well they remember the appearance of those replicas.
What is the Math or Science equivalent of saying: "I can read"?
I have an answer for history: I can tell the story, in five minutes, of how the generations before me brought me to this place and time. I think that creates a fundamental enough experience that no one would forget the core narrative. But it's impossible to test that theory today; because that is certainly not how we teach history.
I don't have one for Math, which is an enormous, undulating bundle of concepts and skills. One can construct simple, individual active sentences around math fragments: I can add. I can subtract. But you can't practice all of them in any systemic way.
By contrast, you can sit down with a book and both practice reading comprehensively and enjoy yourself. Math, by contrast, overwhelms with its sheer volume.
So what, exactly, are high stakes Math tests testing?
What is Algebra?
1. a generalization of arithmetic in which letters representing numbers are combined according to the rules of arithmetic.
2. any of various systems or branches of mathematics or logic concerned with the properties and relationships of abstract entities (such as complex numbers, matrices, sets, vectors, groups, rings, or fields) manipulated in symbolic form under operations often analogous to those of arithmetic — compare
Now, try to define lasting mastery of that.
What's I can Algebra? Again, not an abstract question. Because if you can't Algebra, according to a bad test, the state of Florida will withhold, at 18, your shot at a shrinking target of a middle class existence. If you can't Algebra at 18, Florida wants it to have profound negative effect on your life at 50, which means it has profound negative on your children, etc.
Moreover, if you're not an engineer or professional mathematician, I wager that you will find precious little applicability for the mechanics of undefinable Algebra in your life and work. (And I wonder about a lot of the different disciplines of engineering. If you're an engineer, how much do your mechanical algebraic skills affect your work?)
Outside of the concept of identifying and solving problems with variables, a concept that extends so far beyond Algebra that one does not even need Algebra to inculcate it, I have found nothing handy in my life that comes from my study of Algebra -- or any other highly conceptual math -- as a kid.
Other than reversing the order of Geometry and Algebra II, I took the same "college track" of math classes my son is taking: Algebra 1 (8th); Algebra II (9th); Geometry (10); Analytic Geometry and Trigonometry (11th); Calculus (12th). And then college Calculus at an elite college, where it was horribly taught by a professor speaking English as a very second language.
I managed to get by with decent, even good, grades, except in college; but I only enjoyed Geometry. And it's the only Math subject from which I retained some useful abilities, specifically how the measurements of circles and triangles work. I also enjoyed the elegance of proofs, although I could not, today, go back and set one up and work through it.
Today, I would trade my entire "college track" Math education for an ongoing Statistics class throughout my high school years. Other than very, very basic conceptual ideas about solving problems with an unidentified variable, I use absolutely nothing related to Algebra in my life or work.
I use basic statistics constantly. (Not even standard deviation-level statistics. More basic.) Over the years, because I use and practice them, I have taught myself statistical rudiments and effectively applied them to a number of real world concepts. I'm probably proudest of designing a statistical analysis of school test score performance a few years ago that argued wealthy schools of all types generally underachieve their predicted scores on standardized tests -- and that the opposite is true for poor schools. (Unfortunately, underlying changes in how we calculate the relative wealth of our schools make that analysis essentially impossible today.)
When I, or others, point out that Algebra -- in all but its most basic concepts -- is utterly useless to the lives of most people, I hear a ready retort: don't worry so much about how you'll apply it in your life. It's an important mental exercise, one that challenges your mind think deeply about solving problems.
I happen to agree with that.
I do think Algebra has value in that puzzle-based, problem-based form. But that's not remotely how it's taught and tested. You can't meaningfully test the static knowledge of an exercise. It's taught as if it's a static thing, a container that you can fill with bits of "knowledge," otherwise known as standards.
It's taught as a race to cover standards with no lasting human value. It's then ruthlessly tested with the goal of locking some number kids out of any shot of a middle class existence. No sane or decent country would deny people a high school diploma over misapplied, volume-based testing of a mental exercise that can't even be meaningfully defined. Madness.
Imagine applying that to other virtuous exercises.
The point of Yoga is to Yoga. It's not to demonstrate your knowledge of Yoga under pain of personal destruction. So, if you are one of those "Algebra is an exercise" people, it's time to invest some political capital in your idea. Because our government's approach to Algebra is killing it. And us.
Memorized Math as a human filter
This piece is, in some ways, a long overdue response to Paul Cottle, a Physics professor at Florida State University. He and I had a fun Twitter exchange a few months back.
And he wrote a very useful and thoughtful "letter" to me on his blog. Read the whole thing here.
I won't try to summarize all of it. He gives good context. But I think, in a sense, he's focused on the top of the math, science, etc. pyramid and trying to make that pinnacle more accessible to the wider swaths beneath it. I'm focused on the broader swaths -- and how to make Math education more useful to them without destroying their futures.
Our points-of-view are both different and reconcilable, I believe. They meet each other -- and find the problem to work through together in the following paragraph, particularly the last sentence.
I used to believe in making policy at the state or local level to nudge students into taking higher level math and science courses (chemistry, physics, precalculus, calculus) in high school. I have given up on that because there is now plenty of evidence that it doesn’t work. Florida’s SB 4, which was signed into law in 2010 and would have required Algebra 2 and “chemistry or physics” for graduation, was repealed before it ever went into effect. More dramatically, the Texas “4×4” graduation plan basically required every student to take Algebra 2 and physics. And the thing is that it worked for a decade: Graduation rates and mathematics achievement went up. What’s not to like? But it was repealed in 2013, anyway. Parents, employers and teachers had never bought in, even with all of the success.
Very quickly: graduation rates everywhere are largely fraudulent and meaningless as measures of educational value.
Injecting a static Algebra 1 test into this fraud and meaninglessness as a condition of getting a shot at a middle class life is absurd - especially if it's primarily a "mental exercise." Yet, every year, a sizable cadre of kids (more than 5,000 statewide) receive "certificates of completion," not diplomas, because they can't pass a high stakes test of no human value about a mental exercise.
So when Paul says "requirements," I hear "lasting life consequences."
And as I discovered in college Calculus, at an extremely expensive and prestigious private college, it's hard for even brilliant people to teach this stuff well enough to justify punishing kids in a lasting way for test performance.
I think Paul's long-term solution to this is widely differential pay for teachers, a thing I could consider if we increased the base pay for everyone significantly. It's a discussion for a different post.
There are many forms of rigorous mental exercise. Algebra has no monopoly.
Moreover, I've been hearing about remedial math from college professors for a long time, since long before the 2013 demarcation line Paul cites. So, I also must question the "success" to which he refers.
And I would argue that parents, employers, and teachers had very rational, very defensible reasons for not buying in, especially because Algebra is not day-to-day relevant to the lives of the vast majority of students, parents, teachers, and employers.
But that's not at all the same as saying that math and science don't consist of important disciplines, mental exercises, and empirical questions that all students should engage. It's the apparatus and consequences of engagement that I believe we are getting catastrophically wrong. And I think we can build an apparatus that does so much better.
Contrary to Paul, I don't think we've ever tried to "nudge" kids into engaging higher level math and science. We've bludgeoned and punished them. Let's actually try nudging. Let's think about how to encourage them -- and ease their fear and intimidation.
And let's consider that liberal arts, which we have essentially stigmatized as irrelevant and useless for a generation, is in fact full of important mental exercises that build lasting capability and human judgment. Those all important "soft skills" you hear about these days from employers have another, ancient name: humanities.
Perhaps my focus on definition and word-meaning and human consequences in math reflects my verbally-oriented, humanity-focused liberal arts education. Perhaps it reflects my insistence that we inject human morality into all of these discussions.
But when I look at the sentence When he says "requirements," I hear "lasting life consequences,"I also see a mental exercise that looks a bit like the mental exercise often described to me by the Algebra-is-a-mental-exercise people.
Am I wrong?