Code cracked

How algebra finally made sense after sixty

IDON’T SEE HOW IT CAN HARM ME NOW TO T REVEAL that I only passed math in high school because I cheated. I could add and subtract and multiply and divide, but I entered the wilderness when words became equations. On test days I sat beside smart boys and girls whose handwriting I could read and divided my attention between his or her desk and the teacher’s eyes. To pass Algebra II, I copied a term paper and nearly got caught. By then I was going to a boys’ school, and it gives me pause to think that I might have been kicked out and had to begin a different life, knowing different people, having different experiences, and eventually erasing the person I am now.

When I read Memories, Dreams, Reflections, I felt a kinship with Carl Jung, who described math class as “sheer terror and torture”, since he was “amathematikos ”, which means something like non-mathematical. I am by nature a self-improver. I have read Gibbon, I have read Proust. I read the Old and New Testaments and most of Shakespeare. I studied French. I have meditated. I jogged. I learned to draw, using the right side of my brain. A few years ago I decided to see if I could learn simple math, adolescent math, what in the 18th century was called pure mathematics: algebra, geometry and calculus. I didn’t understand why it had been so hard. Had I just fallen behind and never caught up? Was I not smart enough? Was I somehow unfitted to learn a logical, complex and systematised discipline? Or was the capacity to learn math like any other attribute, talent for music, say? Instead of tone deaf, was I math deaf? And if I wasn’t and could correct this deficiency, what might I be capable of at 65 that I hadn’t been capable of before? I pictured mathematics as a landscape and myself as if contemplating a journey from which I might return like Marco Polo, having seen strange sights and with undreamt-of memories.

I could have taken a class, but I had already failed math in a class. Also, I didn’t want to be subject to the anxiety of keeping up with a class or slowing one down because I had my hand in the air all the time. I didn’t want a class for older people, because I didn’t want to be talked down to – and more cheerfully than in usual life, the way nurses and flight attendants talk to you. I could have sat in a class of low-achievers, a remedial class, but they aren’t easy to find. I arranged to occupy a chair one afternoon in an algebra class at my old school, where 12-year-olds ran rings around me. The teacher assigned problems in groups of five, and by the time I had finished the first problem they had finished all of them correctly. They were polite about it, and winning in the pleasure they took in competing with one another, but it was startling to note how much faster they moved than I did. I felt as if we were two different species.

Having skipped me, the talent for math concentrated extravagantly in one of my nieces, Amie Wilkinson, a professor at the University of Chicago, and I figured she could teach me.

“How do you think this will go?” I asked Amie.

“If I had to guess, I would say you will probably overthink.”

“How so?”

“X is a useful thing. I can solve for it – I can manipulate it – and I can hear you say, ‘What does that mean?’”

“Do I whine like that,” I thought, then I said: “What does it mean?”

“It’s a symbol that stands for what you want it to stand for.”

“What if I don’t know what I want it to stand for?” “See, this is what I’m talking about.”

“Well, wait, that’s …”

“Here is some advice,” she said firmly. “I get it that you try to put things into a framework that you can understand. That’s fine, but at first, until you become comfortable with the formal manipulation, you have to be like a child.”

To prepare for our meeting, Amie suggested that I read Algebra for Dummies, which I had hardly begun when it was borne in on me that it didn’t matter who it was for, it was still algebra. Reading the book, I am surprised to find that I recall almost nothing of algebra. I had got lost so quickly that very little had made an impression. I can still recite the prologue to the Canterbury Tales in Middle English, which I was required to learn as a senior in high school. I remember “kingdom, phylum, class, order, family, genus, species”. And that in 585BC, Thales predicted an eclipse of the sun. With algebra, I come up empty.

When I thought I had read a sufficient amount, I went to Chicago to see Amie. I sat beside her on a couch in her living room. I held my pencil and notebook ready. My manner was like that of the novice on his first day in the monastery, poised to have the head monk reveal how to find God. She said: “I’m not sure where to start.” I had been expecting her to say something like: “There’s a train in Omaha heading for Dallas and leaving at three in the afternoon.” Instead, we sat silently. A dog barked. I smiled weakly.

THERE IS A BELIEF AMONG CERTAIN ACADEMICS that a subject is less efficiently learned from an adept than from someone who is studying it or has just finished studying it. The adept’s long acquaintance makes it difficult for him or her to see the subject in its simpler terms or to appreciate what it is like to approach the subject as a greenhorn. As I sat uneasily beside Amie, it dawned on me that I was asking a mathematician with a trophy case whose standing is international to teach me math that she had learned nearly half a century earlier as a precocious child and hadn’t used since. Furthermore, she had for the most part embraced it intuitively and then layered upon it many other practices, explorations and diversions. Her learning had a kind of family tree of associations, and all I had was what I had picked up piecemeal in a few weeks of study. What I might have said to her of the difficulty I was having was: “Pretend you were a child receiving this information for the first time. Can you remember how you heard it so it was sensible to you?”

I am having to learn how to learn. In school they expect you to learn, but they don’t teach you how to lealearn

A further complication developed, which is that what is difficult for me had not been difficult for her, and I don’t think she could see why I had such trouble learning what she had found simple. “How do you think you would have thought about this if you hadn’t been able to think of it as you had,” is the kind of question I would have had to ask, and being philosophical more than practical, it isn’t a discussion that would have solved my difficulties. I might have learned something about her, but not likely anything about math. In On Proof and Progress in Mathematics, William Thurston writes: “The transfer of understanding from one person to another is not automatic. It is hard and tricky.” We had been working together in a halting way for several weeks when I realised that I was going to have to learn a lot of this on my own.

We don’t often encounter the limits of our intelligence, but the way I struggled with algebra sometimes made me wonder if I was finding my own. At such times I felt myself to be a poorly equipped version of human possibility, sort of a discard. I was also almost daily reminded of how some things needed to be learned more slowly. Meanwhile, I was harassed by my upbringing to believe that I had to work quickly; any half-smart person could work out a problem given sufficient time. I found these attitudes difficult to combine.

Sometimes I realised that I was talking to different parts of myself, and the exchange was not polite. Occasionally,

I got good at operations that were hard at first. This happened with factoring, a process in algebra of simplifying expressions, and with expanding, which is the opposite of factoring. The axioms of arithmetic imply that when you expand (a + b) 2, for example, you get a2 + 2ab + b2 in the following way: (a + b) 2 is equal to (a + b)(a + b).

Each term in one parentheses multiplies the terms in the other: a × a = a2; a × b = ab; b × a = ab; b × b = b 2. Combining the terms, a 2 + ab + ab + b 2 = a2 + 2ab + b 2. In a similar way, a2 – b2, a squared number subtracted from another squared number, called a difference of squares, becomes (a – b) (a + b), which becomes a2 + ab – ab – b2, which is a2 – b2. Simple, but I really liked it.

As the formulas became more complicated, there were more steps, each of which followed from the one before it, so that in addition to finding the answer, there was the pleasure of enacting a procedure properly, plus no textbook skipped the steps. Each time I turned a page and saw more factoring, I was pleased. It was like being good at spelling and wanting to be asked more words. Accompanying my pleasure, though, was a voice saying: “Listen, Slick, this is algebra for kids. We can throw problems at you, you won’t even know what they mean.”

Sometimes I dreamed that there were numbers falling from the sky into chasms I couldn’t see the bottom of. As I progressed, my eye progressed, and more than solving algebra problems by grasping their design, I became more clever at reading questions. To do better, though, I had to become vigilant. For someone who thought that there were shortcuts and faster passageways to learning, this was unwelcome. I had never understood that learning needs to be done patiently. One can be impatient to learn or for learning in general, but that is a matter of temperament. I am having to learn how to learn. In school they expect you to learn, but they don’t teach you how to learn, at least they didn’t in my childhood.

I AM ACCUSTOMED TO REMEMBERING WHAT I HEAR AND BEING ABLE TO DRAW ON IT. Learning algebra requires a secondary use of information, though, a sorting and referencing, a repetition of experience, so that it actually is experience. With algebra I’m not simply collecting information, I am having to classify and comprehend it. We do this naturally as children in classrooms, partly because the distant future seems as if it will never arrive, but it is a different matter to be older and feel that one’s capital of time is remorselessly diminishing. Such a consideration adds a complicating haste and impatience.

The ability to learn mathematics is thought to decline around 40, when the brain begins slowing its handling of procedural operations such as calculating. Older people learn and forget at roughly

the same pace that younger people do, but calculating takes an older person twice as long. In the paper Acquiring Skill at Mental Calculation in Adulthood, Neil Charness and Jamie Campbell say that middle-aged people perform as older ones do, but if they practise, they perform more as younger people do. If speed is valued more than accuracy, the decline in ability is obvious. If accuracy is valued more than speed, the decline is less obvious and maybe not even very pronounced. Younger people tend to read faster than older people. Older people tend to remember more of what they’ve read.

From brain scans it appears that older people engage more of their faculties in solving a problem than young people do. The scaffolding theory of ageing and cognition says that brains respond to declines by recruiting assistance – that is, by replacing a response typically dedicated to a single area with a pattern of layered responses involving several areas. “Harold” is an acronym for “hemispheric asymmetry reduction in older adults”, a form of brain plasticity. I know about it from the research article Creativity and Ageing by Gene Cohen. Cohen says that the brains of older people may enlist areas that usually have one function to collaborate with another function, which is called bilateralisation. Cohen likens it to the brain’s moving, perhaps in a compensatory way, “to an all-wheel drive”.

CAROL ARO D RYFF AT THE UNIVERSITY OF WISCONSIN’S CON INSTITUTE OF AGEING told me about abo stereotype embodiment theory, which was proposed by the Yale psychologist Becca Levy. It says that the culture presents older people as moving slowly, being hard of hearing, talking too loud, and unable to read small print. prin These depictions are funny when we’re young; you then we grow old and enact them, and they the undermine a person’s sense of wellbeing. “There are certain fields where you get better with age, though,” Ryff told me. “You’re not going to have a 22-year-old 22 year old wunderkind psychotherapist. Most of Freud’s brilliant theories didn’t arrive until his 50s.” I told Ryff that I was trying to learn math, and that I had a math allergy. “Someone with math anxiety, later in life, with a different perspective can really shine and discover something new,” she said. “It’s incredibly healthy for the brain as well.”

I felt I was beginning to change, to a degree, perhaps only in a cursory way, but I also felt, superstitiously, that to acknowledge it might be prideful

DIVIDING THE FRACTION 7/2 BY 2, I confused the properties of exponents, and thought the product is 7, since 7 × 2 = 14 and 14/2 = 7, when in fact the answer is 7/4, since dividing a fraction by 2 is the same as multiplying by 1/2, but I got the answer wrong and got angry at math and called Amie, and she wouldn’t talk to me until I calmed down. She wasn’t always calm, either. Once I heard Benson, her husband, in the background say: “Why are you yelling at him?” When I had worn Amie’s patience too thin, I would call Deane Yang, my friend who is a mathematics professor at NYU.

“The way you remember procedures is you remember why,” he said.

“Because?”

“Because people learn math as a collection of procedures,” he said. “When things get difficult, they’re lost, and math becomes religion class. The teacher says what’s right and wrong, and for all you know math came out of the sky, and some prophet told you how to do it, and it’s just blind belief then. The goal is to take on questions that appear to be complicated, and to recognise that a complicated question can be broken down into simpler questions, some of which can be answered independently of one another.”

“With math you have to be very, very disciplined,” Deane said. “Normally with algebra, you’re trying to make something complicated simpler, but often, temporarily, you have to make it more complicated. The only way to be properly disciplined is to remember exactly what you’re allowed to do, and what you’re not allowed to do. You have to write everything down, line by line. Math is painstaking.” These remarks had an almost Zen-like forcefulness for me. They were both abstract and practical, they spoke to my distress, and for a while things got better.

I finished algebra plagued by the feeling that I had to get every problem right. I had started hopefully and been throttled. What I had wished for was to see algebra as rational and cohesive, and therefore benign, so that I could dispose of the mystery it had left me with. If I were able to do that, I would have made use of ways of thinking that challenged me to expand my intellect – my capacity for regarding problems whose solutions require the management of symbols, something I had never been good at.

The enlargement of one’s intellectual reach isn’t the kind of circumstance a person can identify empirically. One can only on sense it about oneself. I felt I was beginning to change, to a degree, perhaps only in a cursory way, but I also felt, superstitiously, that to acknowledge it might be prideful, which wh might lead to it being revoked by whatever agency it is that t lurks inside superstitious moral attitudes. Anyway, I finished algebra, I came to the end of the textbook. It had taken five months, not six weeks. I had learned things, though, I had some new skills, even if rudimentary ones. I could cou do things I hadn’t been able to do, and I was pleased. The Th accomplishment was not substantial, but it was my own, and I had worked for it. I raised a private glass to myself and said, “Well done”, in the middle of an afternoon – then I went on to geometry •