I realize that I don’t post many stories about my positive experiences with students. Part of it is because those experiences don’t require the sort of catharsis that the horror storiesdemand. Part of it is that my darkly comical style of writing doesn’t lend itself to expositions of, like, students actually learning stuff from me. And part of it is because sometimes, the success stories depress me even more than the failures.
I’ve been tutoring a high school kid for the past two months. The kid’s in grade 12; when I met him, he was doing math at a grade two or three level. This is not an exaggeration: he couldn’t add or multiply single-digit numbers without a calculator. And this wasn’t just rustiness, as this inability extended to not being able to compute things like 6+0, 5*1, or 3*0. In other words, he didn’t know what numbers were. Not surprisingly, he couldn’t solve linear equations, add fractions, or make heads or tails of the most simple word problem.
I met with him every other day, two hours at a time. And, to his credit, what he lacked in mathematical skill, he more than made up for in persistence. He worked diligently, if not terribly successfully, on his homework. We spent a lot of time on the basics – fractions, simple algebra, the meaning of equations. We also spent a lot of time – far, far more than I’d have liked – on how to use the fucking graphing calculator to perform tasks that every student should know how to do with a pencil and paper.
He’s two thirds of the way through the course now, and he’s pulling an A-minus.
And he’s learned a lot in this class, and I’m glad that his effort will almost certainly earn him the C he needs by the end of October. But little of what he knows falls into the category of “mathematics”; the bulk falls under the umbrella of “tricks that will get a student through a grade 12 math class in BC.” He’s vaguely familiar with basic algebra now, but he still falters when simplifying an expression (”Am I allowed to cancel out the x’s in (x+5)/x?”).
He cannot immediately identify which methods to use in solving an equation – he’ll use the quadratic formula if someone tells him he’s dealing with a quadratic equation, but he needs that push. Meanwhile, he’s a whiz at using his graphing calculator: he’s mastered all of the fancy features, and can program all of the rules into it. (He’s managed to fit, in the calculator’s memory, examples of every type of question he might see, along with solutions.) After each test, he purges both his calculator’s memory and his own of everything we’d studied in the previous chapter.
Two weeks ago we wrapped up the trig unit, which vexed him even more than the previous four chapters had. Following some introspection, he was able to identify the source of his frustration: why, he wondered, did those bastard curriculum designers make him do this shit? Didn’t they know that the TI-83+ could neither do proofs, nor provide exact values of sin (pi/3) and the like?
During one particularly trying session in which I was explaining, for the third time, how to find those values, he declared that he wasn’t going to learn “that triangle shit”, and I found myself, for the first time ever, raising my voice with a student. He relented. He got an A on the trig test. The next week, reminiscing, he asked, “trig – we did that already, right? Was that the stuff with the logs?”
Last week we started on the combinatorics chapter, and I braced myself for the damage. Combinatorics, unlike most of the rest of the course, requires some creativity: formulas are few, and variations on themes are many. Every question must be read carefully, and even the simplest ones require students to think about how to set them up.
However, my student took to this section surprisingly well, and found it to be a lot less stressful than the others. He even asked me some questions that were related to the material, and not just to how to pass the next test.
The last section of the combinatorics unit covered the Binomial Theorem, which provides a shortcut for expanding expressions of the form (a+b)^n. I always thought that this material was the easiest part of the combinatorics chapter: it’s completely algorithmic, and demands no creativity.
It does, however, demand a knowledge of basic algebra.
“Here’s what we’re going to do in this section,” I explained. “We’re going to find a shortcut for expanding things like (a+b)^n – n is a whole number.”
“Which whole number?”
Sigh. We’ve been through this, many times. He always gets tripped up by questions that require him to generalize anything, even slightly. Has to do with skimming over the directions, and failing to understand that variables…can vary. Last month’s lesson didn’t sink in, apparently.
I explained it again. “N can be any whole number,” I said. “We’re going to introduce a formula that will let us expand (a+b)^1, (a+b)^2, (a+b)^3…and (a+b)^n for any whole number n.”
And away we went. We had worked out the first several lines of Pascal’s Triangle on the previous page, and I made sure that that was accessible as I had him expand the powers of binomials the long way.
He started having trouble with (a+b)^0. “Zero?” he asked. Even though he’d raised things to powers of zero in the chapter on geometric series. And the one on exponents. And a few others.
“What’s anything raised to the power of zero?”
He reached for his calculator and tried a few values. Frustrating, but at least he knew that he could figure out the answer by trying some values. He hadn’t known that in July. “One,” he declared.
The expansion of (a+b)^1 proceeded without incident. He was just as quick to give an answer for the next one: “(a+b)^2 = a^2+b^2,” he said.
We’d gone over this one a dozen times in the previous month. He’d made that mistake a dozen times. Each time we went over the basic rules of algebra, and tried plugging in numbers for a and b that showed that (a+b)^2 doesn’t always equal the same thing as a^2+b^2. But that was last month. He was supposed to know this still?
And it wasn’t over. You mean ab and ba are the same? So we can collect them? And we can’tcollect ab and a^2? Why not?
Anyway, enough. My point is: in the BC high school math metric, this is A-minus work. A good graphing calculator and a mediocre short-term memory are sufficient to achieve excellence in high school math classes; most students will take the path of least resistence and develop little else. Grades of C or higher are required for the college program this kid’s trying to get into.
Grades of D in this course are sufficient to gain entry into the precalculus class I taught at Island U last year. If a failure to grasp basic algebra doesn’t stand in the way of getting an A-minus, then what on earth does one have to know in order to get a C? Or a D? Why not just eliminate the middle man and admit anyone into a college math class, rather than wasting everyone’s time on this extended TI-83+ how-to seminar?
This sort of thing is part of why I usually have little sympathy for students who claim that, no, really, they do know the material, it’s just that they do badly on tests. In my experience, it’s more frequently the opposite: they do far better on tests than their actual understanding of the material reflects. And then they sink like stones when they take a college class with me, in which I design the tests and don’t allow them to use their graphing calculators.
Well, not all of them sink. I had a student last year, who earned a C on the first test and then quickly adapted to my expectations. He had the aptitude to do so; it’s just that he’d never been required to use it to full capacity. He pulled an A-minus on the final exam, and wrote me a note at the end. I transcribed it, and I go back and read it every now and again when I’m particularly frustrated:
I hope I did as well on this exam as I think I did. I know it took me awhile to get going this term, but I worked really hard and I finally feel like I get this stuff. I always used to do well in math class in high school, but you actually made me think. Even though I got good marks in math class before, I feel like this is the first time I really learned math.
I guess that’s success. I expect a thank-you from my current tutee, who certainly would not have achieved the required C without me, and who will almost certainly get or surpass it next month. But I doubt I’ll feel all that good about it.