Learning Math

Jessica Lahey’s In Defense of Algebra, in which she describes her adult triumph over math, reminded me of my own experiences learning math and how they might be relevant. This is long, but if I make what I did sound too easy, people will get the wrong idea.

I struggled in math during high school, but was simply too clueless to quit. I got As and Bs algebra in 8th grade, got Cs and Ds in the subsequent courses until senior year, when I held onto a B the entire year in AP Calculus and, in what remains one of the great academic shockers of my life, passed the AP Calc test with a 3. Many years later, I took the GRE for my first master’s. I spent a month slogging through math, relearning enough of it to get by, and was very pleased with my 650 quant score, which was the 65th percentile on the GRE. Good, but not great, which is pretty much how I did on the SAT many years earlier. (Verbal, which I spent no time on, was 790).

My math turnaround began after I started grad school, when my son was failing geometry. After reading his book and working dozens of problems, I was able to help him (he went on to pass both the AP Stats and Calc AB tests, and is stronger in college math than I am). Needing a part time job for grad school, I auditioned for a job at Kaplan.

I originally hired on to teach GRE classes, but within a month I was working close to 40 hours a week teaching the high school tests. Two months later, I was teaching Math 1c and Math 2c, which was ridiculous. I had to learn the SAT math by rote at first, and the SAT Subject math tests required trig and second year algebra. I protested to no avail, and my manager’s decision was prescient—within a few months, parents were emailing me for private homework tutoring in high school math. It turns out I’m very good at explaining things; when I didn’t know how to do a problem in the early days, which was often, I’d go look it up or ask a student who did understand the problem. Ah, that’s the part I missed! Anyone else do that? I could see students nodding. I made the process very transparent, showed students where the glitches in my comprehension were, helped them find their own glitches and, it turns out, students would rather be tutored by someone who knows where the understanding glitches are.

I told parents I didn’t have a clue what a log was if it wasn’t in an Ingalls story, but even after raising my rates as an attempt to scare them off, I was getting lots of work. Still, for my own peace of mind, I decided I needed to learn more math instead of literally learning it while teaching others.

I started with test math, my strength. The REA and Kaplan test prep books served as my educational foundation. Between the two books, I learned how to recognize the subject and how to solve a number of common problems. When I was tutoring students, I would recognize the type of problem it was. Then, using their textbook, I’d help them work problems by example and explain. In explaining the math, I learned a great deal. My students learned and got improved grades, more confidence, and better test scores. Me, I kept getting more clients.

After four years of this, I had an excellent understanding of high school math through pre-calc, and routinely scored 800 when taking practice Math 2c tests (which I did for the first few years to keep me alert to weak spots). I not only knew the material cold, for the most part, but was by this point extremely familiar with the curriculum and sequencing for pre-algebra through algebra II/trig, and somewhat familiar with the same for math analysis (pre-calc) and calculus.

Seven years after my first GRE and untold decades after high school, I aced two of the three teacher math credential tests, and passed the third, in calc, with what can be called a gentleman’s C—the conceptual questions, the trig and the math history questions pulled me through. I scheduled my ed school GRE with two days’ lead, and got an 800 on the quant—by that time I was getting 800s on the practice test without a pencil, sitting around the Kaplan office with 15 minutes to kill. (Verbal: 780, but I was distracted and finished in 12 minutes.)

As I was studying for the calc credential test (which I took twice), I suddenly had an epiphany about why I was now able to learn math, when I’d done so poorly in high school, and for this, boys and girls, I must go back even further, back to the dark old ages of the mainframe and the earliest days of my previous career:

I was unfocused after college, having done well in English lit classes and nothing else. I started as a data entry clerk, but got a contract at IBM using a mainframe product that everyone in its customer base hated, but had to use. While entering the data, I noticed a few short cuts, asked for a manual, and began customizing the product in ways very few people knew was possible. The boss was impressed and made me responsible for product demonstrations, showing my work and explaining how the product could be customized. I got a job from one of the companies that came to the demo, a major brokerage firm. It was my first real job after five years of temping in admin jobs, during and after college. (Irony alert: I’d taken, and nearly flunked, one computer programming course in college and was convinced it wasn’t the career for me. I was three years into my new job before I felt comfortable saying I was an applications programmer, much less a computer programmer.)

And the first day I got to my new job, my boss said to me, “So I need to put you on a different project first. Our systems management application needs to be installed on the NYC machine.We have this automated process that moves the source code out of Panvalet, automatically builds the compile JCL based on the program—you know, if it’s CICS or batch, IDMS or DB2, and so on, and compiles it, linkrefs it, and installs it in the right library. It’s already working here; you just need to transfer all the code, change the libraries, and so on. You know CICS, right?”

“No.”

“Oh, it’s a transaction server environment. We mostly use COBOL here, but we’ve got some assembler routines. You know COBOL or assembler? No problem, I’ll sign you up for a class. The code itself is mostly ISPF calls with EXEC, although I’d like to upgrade to REXX.”

It’s not just that I didn’t know COBOL, and had no idea what CICS was. It’s that I didn’t know what compile meant, or source code, or batch, or JCL, much less IDMS or DB2. And I didn’t have the foggiest clue what REXX or EXEC was, and ISPF to me was an application, not something I programmed with.

Baby, I brought that motherf***er in on time. Two and a half months. I got a bonus, too, because by the time it went live, my director had figured out a small fraction of how much I didn’t know, and was extremely impressed. She never grasped the sum total of my ignorance, thank god. Nor did she ever realize that I still didn’t know what the difference between CICS and batch was and didn’t realize that (at the time) a program couldn’t be both, only vaguely understood that “compile” meant translating code I could read into weird symbols I couldn’t read but presumably the computer could, never did learn COBOL, and only vaguely understood what JCL was or what it did. All that was in my future. The only thing I was pretty confident about after those two and a half months was that I was pretty darn good at EXEC and ISPF dialogs, and that these things weren’t what the brokerage products ran on.

So my epiphany was this: Working with computers had taught me how I learned. How I learned. Which was not like most people. When books don’t work as a learning tool, then I have to learn by a particular type of doing. Explanations won’t help. Learning in a vacuum won’t help. I need to learn by trial and error. And then, I learn like Wile E. Coyote traverses the desert; I just keep on going until something blows up in my face. Go this way? Boom! Okay, that way doesn’t work. File it away. Go that way? Two steps, yes, then BOOM! Okay, the two steps, file away, then don’t go that way because BOOM! how about this way? Tiptoe, tiptoe, try this, ha! It worked! Done. On to the next. Make sense of the chaos, bit by bit, understanding the rules by the reaction.

When I’m learning something, I neither know nor care about why. Understanding will usually come. So just as I ultimately understood CICS only several months after I started making changes to a mission critical CICS transaction, I didn’t bother with understanding what, exactly, trigonometry was. Two years after I first learned how to work trig problems, I read that trigonometry was the study of the ratio of right triangle sides, and I was like Holy Crap, that’s exactly what trig is. What a trip. One day soon I’ll internalize the fundamental theorem of algebra, but give me time. If not, I’ll wave the dead chicken over the problem because that’s what worked the last time. And it usually works again. (Note: Ironically, as a math teacher, I am big on explaining why, but that’s because I’ve realized that most people aren’t like me.)

Of course, theory, whether it be math or computers, is usually beyond me. And yet, in both math and computers, I am capable of occasional insights that please actual mathematicians and computer scientists, even though most of the time I don’t care, as they are working on things that I find utterly incomprehensible.

Why did I struggle with math in high school? The usual reasons don’t apply. I was an A student, and remained one in English and history. Unlike Jessica, I never felt labelled, nor did I give up. I had excellent math teachers, all of whom knew that my intellect was considerable and took the time to reach out—each one sat me down at some point and asked why I wasn’t doing better, given my obvious brains.

I just know that some people are going to read my post, as they read Jessica Lahey’s, and conclude that, by golly, we prove that anyone can learn math, and that labelling kids based on early progress is cruel and wrong and demoralizing. Lahey herself clearly holds this position.

Well, no. Lahey and I are both extremely bright and I know I say this a lot, but that’s because people persist in ignoring the relationship between “smart” and “academic achievement”. Lahey clearly has excellent verbal skills, strong at writing and foreign language (she’s a Latin teacher at an elite middle school), whereas I’m a hybrid who, in addition to excellent verbal skills, tested high on every computer aptitude test that came out back when I was in college. On the other hand, I can’t speak any foreign languages, and I suspect Lahey isn’t as strong on logic and pattern recognition, which is why she needed an algebra teacher to get through first year algebra, whereas I self-taught myself the entire high school curriculum.

What Lahey and I both demonstrate is that it’s possible to be well above average in smarts, yet still struggle in math when later experience proves that we were entirely capable of grasping it.

Why, then, does an otherwise smart person struggle with math?

I have a theory, involving my layman’s understanding of IQ, which I’ll go into briefly.

Two visual aids to categorizing or measuring intelligence: Wechsler Adult Intelligence Scale subscores and subtests and Cattell-Horn-Carroll theory. In both, you can see what most people know on a casual level: intelligence has a verbal component and a visual/spatial component (known as performance in Wechsler). Logic seems to cut across the categories. It’s not terribly controversial to point out that advanced math, even that found in high school, requires more visual spatial and logic ability. I don’t know, specifically, how my intelligence maps to these categories. But I’ve always known that my verbal abilities were very high, my pattern recognition and decision processing equally so, and my visual-spatial relatively weak.

Imagine smart kids who has really strong verbal skills but unknown weaknesses in either logic or visual spatial abilities. These kids would coast easily through elementary school, where the skills needed are almost exclusively verbal—reading and arithmetic. By the end of of 8th grade, they’re bored out of their minds. Most of elementary school is time spent teaching them things they already know and developing social skills.

So for 8 years, this type of smart kid hadn’t ever had to struggle to learn something—in fact, learning itself is pretty alien to smart kids. (This, parents of smart kids, is why you should make sure your kids have to struggle with something—cooking, art, horseback riding, making nice with other people, whatever.)

And then, math. Algebra and beyond can come as a big shock. When school has come easily all your life, it’s hard to even know what “learning” is, much less how it applies to you. I’ve talked to countless people who qualify as Really, Really Smart, and every one of them has agreed that at some point in their lives, they realized that they had no idea how to learn. Some figured it out in high school or college. Some, like me, didn’t figure it out until they got a job that required them to learn something. Others, like Lahey, had access to a math teacher and went back and learned it because they wanted to fill in a gap that they felt was necessary for parenting.

So the otherwise smart kids who struggled with math did so in part because their particular intelligence was strong on verbal, and lighter on the spatial and (probably) visualization skills that are helpful in math. Plus, math was, to quote the Great Barbie, tough. And these kids had never once faced “tough” in a school subject. Many folded.

This was particularly true in the era before we began demanding higher math of all students (say about a decade or fifteen years ago). Before the 90s, math teachers weren’t held responsible for their students’ failures, and we accepted that not everyone was “good at math”. Kids with strong verbal abilities just took less math—in those days, it didn’t end your hopes of college, even of elite college. It was quite normal to get into an excellent school with a strong performance in history or English with little more than second year algebra on a transcript.

Today, of course, any white or Asian kid who wants to go to an elite college has to have advanced math, so the smart verbal kids who don’t have the requisite math skills have a much stronger incentive to either compensate or fake it, with or without a tutor. Moreover, math teachers these days are far more likely to reward effort over ability, so it’s easier for a student to get As in math by religiously doing homework and extra credit, even if they do poorly on tests. And of course, math teachers are also less likely to dismiss the effort involved in teaching math to those who aren’t necessarily strong in the subject. Thus, for many reasons, smart kids today with primarily verbal skills are less likely to have given up on math and are at least willing to fake it. Many learn to compensate, as I did much later in life, by using their other cognitive abilities to make up for their relative weakness.

(Compensation: Picture a circle inscribed in a square. Can you estimate the ratio of the circle’s area to the square’s? Or would you, like me, be completely incapable of a reliable guess and so calculate the difference by creating a radius of length r and work it as an algebra problem? I contend that people who can make an accurate estimate and do the algebra have an easier time in math than the people who can only do it with the algebra.)

Back in the day, the kids whose intelligence was strong in the math aspects but weak in verbal got a similar shock when they were expected to write an analytical essay on Hamlet and offer some original insights. Unfortunately, as has been noted before, our history and English curriculum has never recovered from the dumbing down it suffered through in the interests of multi-culti, giving kids who are strong at math little reason to ever struggle to access their verbal abilities.

This has obvious implications for the big algebra controversy kicked off by Hacker. But it involves recognizing that underlying cognitive ability is a huge determinant when deciding who should, or perhaps shouldn’t, take algebra.