Tag Archives: trigonometry

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?”


“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.

Teaching Trig

(No, I am not in Alaska. Yes, I am overly fond of alliteration.)

My trig lessons were going pretty well, when I noticed that a good number of kids were having trouble identifying the opposite or adjacent side of the angle. This always seems trivial when you’re teaching a big, complicated subject. The kid gets SOHCAHTOA, gets the ratio, gets the concept of inverse, and just has a little bit of trouble figuring out which angle is which. Big deal, right? Lots of concepts grasped and a glitch on implementation. But from a testing and demonstrated knowledge standpoint, the implementation is the ball game.

It suddenly dawned on me that I was constantly guiding students on one point, and one point only, despite a lot of time spent explaining different ways to find opposite or adjacent sides. And that this seemed awfully familiar, a memory from teaching this two years ago. I couldn’t assume they’d finally pick it up, or that this was non-trivial. I realized I needed a different approach.

And so, I gave them a Kuta handout, magic markers, and an explanation:

“Suppose I ask you to find the trig ratios for angle A. You guys have shown me you know the ratios of SOHCAHTOA. But how many of you are still not quite sure which is the opposite, the adjacent, or the hypotenuse?” (Half the class had their hands up before I finished the last sentence.)

“Yeah, that’s what I thought. So we’re going to try this. In your notes, create a right triangle like this one. Using a marker, trace the two sides of angle A.” (I can see all the kids, not just the usual worker bees, busily creating triangles and marking angles, which is encouraging. They want this method; it’s clearly a hitch in their understanding they want fixed.)

“Now, here is an important concept. The angle is created by the hypotenuse and the adjacent. The opposite is not a part of the angle.” (I repeat this several times. I can hear several kids already saying “Oh, I get it!”)

“So, which side is NOT a part of angle A, Corinne?”

“B and C.”

“Okay, remember that you describe segments by saying the two points together, like this: BC. And yes, you’re right. Segment BC is NOT a part of angle A. Sandy, what did I just say about a side NOT being being part of the angle?”

“The opposite. You said the opposite is NOT a part of the angle.”

“Awesome. So Marco, which side is NOT a part of the angle and so must be the OPPOSITE?”


“Great. So everyone mark BC as the opposite, like this:”

“Now. You’ve got a foolproof way of finding the opposite. Just trace over the angle, and then label the side you haven’t traced as the opposite. Now, what about the angle itself? What sides form the angle, class?”

“The adjacent and hypotenuse.”

“But how do you tell the difference between them?” asks a student.

“It’s opposite the right angle,” pops in another.

“True, and that’s a really helpful method. But if you’re worried about what “opposite” means, just do this. Take your pencil, start at the right angle. Draw a straight line through the interior, the inside, of the triangle, until you hit a side. That side is the hypotenuse.”

(Again, I can see that all the kids are doing this. It’s really friggin’ awesome to to realize how much the kids wanted to close this loophole and to have a method be so instantly useful.)

“And now, what’s left?”

“The adjacent!” the class choruses.

“There you go. That’s how you identify the sides. Helpful?”

“Yes!” (big response.)

We practiced it several more times and then I let them loose on the handout. I knew I was on the right track when one of my weakest students showed me his entire handout labelled correctly, but told me he wasn’t sure what to do next. We went through SOHCAHTOA and the light dawned. He worked every problem correctly.

From there, the next big teaching challenge was inverse, when I finally let them use calculators. Yes, up to now, they’d been using a trig table. I wanted them to fully internalize the fact that each trig value is a ratio. Their familiarity with the trig table really aided their comprehension of the inverse.

Next up was solving right triangles and figuring out what ratio to use. But this was much easier because they were now close to 100% accurate in identifying sides, so they weren’t compounding errors.

And then, the course-alike quiz:

This quiz is by another math teacher, and much easier than my quizzes. The kids nailed it. The mode–the MODE–was 100%. The mean was 83%. Only four kids failed the test, and one of them failed because he wasn’t trying.

To reward them for their great job, I still graded the quiz on a curve (meaning one of the Fs converted to a D), and weighted the quiz as 100 points instead of 50.

Then, the kids spent two days moving back and forth between the three methods of solving for sides: Pythagorean Theorem, special rights, and trig.

Once, when I was up front reviewing the results, a kid asked “So why can’t I just use trig instead of special rights?”

Another one chimed in. “Yes, I really don’t like the ratios.”

I said, “I’m fine with you using trig. But remember, you are expected to know the trig ratios for special rights, although we don’t call them by the same names. So if you really like using trig, take a little time to memorize certain values.” And then I went through the trig values for sin(30), cos(30), sin or cos(45).

“So just memorize these three, and you’re fine. And of course, you should never forget the tangent of a special right triangle.”

“ONE!” said a good number of the class. Yay.

We’re moving onto polygons, which will allow me to revisit trig when finding areas.

I’m psyched. If only I had a job next year!

Trig Progress

Last week, I mentioned my plans to help my students understand the ratio element in right triangle trig.

It worked! I didn’t let anyone call out the answer; the kids had to discuss it in their groups first. As I walked round the room, I could hear them tussling with the question. In some cases, one or two of the students figured it out and explained it to everyone else; in others, the group “got” it while talking about it.

I gave them triangles with clearly distinguished cosine and sine ratios, and could start to see it click. Then I drew a 30-60-90 and 45-45-90 triangles, asking them which had a tangent of 1. Half the class figured out it was the isosceles right.

Seriously, it’s sinking in. All of the students in both classes were successfully solving for unknown sides by the time we tabled trig to focus on the state tests.

Now, the next big checkpoint: will they all be able to move between solving using trig ratios vs. the Pythagorean theorem vs. special rights. Here’s hoping.

Texting on Tangents

I don’t collect homework. My students take a picture of it and text or email it to me, or they have me take a picture of it. Then I go through and mark the homework done when I have time. In other words, my students have my email and phone number. To date, they have only used the information for good.

Like this text exchange with a freshman geometry student as he sent me an image of his homework, which was a simple right triangle trig task. I’d sketched two triangles, marked an angle on each, gave them three sides (a,b, and c) and asked them the sine, cosine, and tangent for each angle on each triangle.

Student: For both answers, the tan cos sine were the same angles.

Me: Yep! But the ratios were different.

Student: I don’t even know what I was solving for. Was it right?

Me: Does it matter if it’s right if you don’t know what you’re solving for?

Student: Of course it would matter. If it was right I could assume that what I did to come to that answer was correct.

Me: But if you knew what you were doing, you would not have to wonder. You would know why all the angles found for each identity were the same. And you would have an easier time with tests. Do you know what tan(52) means? That’s the big question.

Student: Don’t think you ever told me and when I asked you didn’t give me a straight answer.

Me: You wound me! I did give you a straight answer. I even brought the question up before the class because I thought it was so important and discussed it. But thanks to our conversation, I will revisit it tomorrow. I appreciate your questions and your honesty. Oh, in answer to your question, the tan(52) is the ratio of the opposite leg to an adjacent leg for a 52 degree angle in a right triangle.

When he asked me that question, I took it to the class and asked three students randomly, “what is a tangent”? Two were able to answer that it was a ratio; the third was able to somewhat explain what, exactly a ratio was.

This particular student is one much more interested in getting the right answer than in understanding the question or any part of it, but it’s still telling that he’s just going through the motions.

How do I get them to think of a trig ratio as a value, rather than just a step in a procedure?

So tomorrow, I’m going to post this diagram and question:

I have changed so dramatically on this point since I began teaching that it boggles the mind. I used to say that it was fine to teach the procedure—to teach the context, the meaning, of course, but if they didn’t get the meaning who cared, so long as they got the answer?

Believe me, I’d still hold to that position if the kids would get the answer. But consider the problems that I think any teacher would recognize:

  1. Students use the quadratic formula for everything–and I mean, everything. Give them a linear equation, two four-term polynomials to add, or a rational expression in fraction form and they’d define an a, b, and c, and plug it in. It’s crazy-making.
  2. Half my students still stumble over the area formula for a triangle.
  3. Students either use the Pythagorean Theorem for everything involving a triangle, or nothing.

That’s a short list. Slowly, I realized that kids who can learn and apply the formulas by rote are not the low ability kids, but close to the strongest in the class. The low ability students just seize on any formula they can remember and plug it in—and much of the time, they won’t remember any formula at all.

They feel better with formulas. When I work with students individually, I’ll prompt them for explanations of their understanding, while they roll their eyes and demand the formula so they can just plug it in. I’ve shut that down by telling them that they’ve been taught the formulas for the past four years, so if they don’t remember it by now why should I waste my time? (Yes, I’m a bit brutal.)

Of course, I teach the formulas. I love algorithms. I often outline methods step by step. I want them to be able to solve problems with the algorithms. But I remind them how useless the formulas can be by asking them questions that require understanding, not just answers.

I’d like to think this is helping their understanding and their long-term math ability. But if it isn’t, at least they are growing uncomfortably aware that the formulas and procedures aren’t as much good as they thought they were unless they take the time to understand the math itself.

And I know I have at least one ex-instructor who’s chortling in his beer at this confession.

I will report back.