You're Not Actually Bad at Math

Perhaps you are. But doubtful. Perhaps you just think you are by some premature appraisal of your ability.

Being “good” or “bad” at anything exists on a sliding scale. I am bad at handling a basketball when guarded by someone who is actually good; even worse at singing; and worse, worse at, say, writing Arabic. But is it really fair to say I’m “bad” at writing Arabic? I wouldn’t say so, because I know nothing about it, and the goodness-badness paradigm, in my view, is predicated on at least minimal knowledge of what the thing you’re evaluating yourself on even is and thus what would constitute “good” or “bad” for it. (I chose Arabic because it doesn’t use the Roman script alphabetic writing system—i.e., your ABCs—that which I do know something about, that which I know how the vowels and consonants and nouns and verbs interact to form words and sentences, so that which you could say I’m bad at writing another language that uses it, like Spanish, which, yes, I am bad at writing.)

I fear I’m losing your attention, but rest assured, the math part is coming.

Knowledge and skill are not the same thing, but usually the former must predate the latter, especially once complexity transcends the most primitive, rudimentary tasks, like running and jumping. One need not know of how the gastrocnemius muscle uses adenosine triphosphate in order to run fast; one needs two strong legs, auspicious genes, a healthy BMI, and a functioning heart and nervous system. But for tasks that require more than the fundamental functions of the human body, having knowledge is immensely helpful, imperative even, in getting skilled. This is true for basically anything, and mathematics is no exception.

There’s multiple ways to look at mathematics and whether someone is good at it. The first way—the popular, perfunctory way—is to look solely at computational ability: how fast, how accurately someone can manipulate numbers, perform arithmetic operations, and work with abstractions using solely their brain. And yes, some people are naturally better at this than others, as some people have a cortex more adept at performing such tasks, just like some people have a musculoskeletal system more adept at generating force. I think we do ourselves no favors in denying such a reality, albeit a politically incorrect concession. But someone can still be “good” at mathematics with inferior ability in the mental math realm. They can be good through the second way: understanding math.

Mathematics has a language, and if you don’t speak a word of it, you’re almost guaranteed to be “bad” at it. Take a polynomial expression, for example. Do you know what a polynomial is? If you can’t define that, even loosely, I’ll bet you don’t know what it means to factor such a thing. Sure, you may remember “foiling” something in high school algebra class, but what does that actually mean (hint: it means something more than the words in the mnemonic)? Did you know it’s often done in order to ascertain the “zeros” of what’s called a “quadratic,” a polynomial of degree two. What’s a polynomial of degree two? What’s a—okay, okay, my point is made. You shouldn’t write yourself off as bad at math if you don’t know a modicum of its language. And that’s no indictment of one’s mental ability. The smartest among us are not born knowing what the hell a polynomial is.

I was motivated to write this article after recently helping a high school student prepare for his SAT. I’ve tutored the mathematics of standardized tests for six years now, and my clients have run the gamut of innate computational ability—the first criterion I mentioned—but almost all are lacking in the second kind, the understanding of the math.

Because this is about math, and because it will clean up this prose a bit, let’s denote the first kind, the innate computational ability, as X, and let’s denote the second kind, understanding math, as Y. A student with only X may do fine on the SAT, as a lot of the material is fairly intuitive, and a student with only Y should do fine on the SAT, as the concepts behind the questions are not overly advanced. But a student with neither X nor Y—well…they need tutoring. As a tutor, my goal is to transform the people who pity themselves on being X’s and turn them into Y’s. As the author of this article, my goal is to do the same.

Remember my earlier babbling about polynomials and quadratics and factoring and this and that? I’m coming back to it to finish my point. (A point, by the way, as defined mathematically, is the notation of an exact location in space. It’s part of the framework of Euclidian geometry, a branch of math named after, you guessed it, Euclid.)

Consider the following problem, which is the same problem I recently did with the aforementioned student:

-3x + y = 6

kx + 2y = 4

In the system of equations above, k is a constant. For what value of k does the system have no solution?

See, unlike taking a math class in school, where the test questions are typically more straightforward—“Solve the system of linear equations,” for instance—getting this question correct requires some background knowledge, knowledge that starts with knowing what those things even are, since they don’t tell us. Those are lines. Those are lines because they are polynomials of the first degree. Those are polynomials of the first degree because the highest exponent power that’s present in each is 1. That implies a constant slope, or an unchanging slope, whereas a curved line, a polynomial of a higher degree, wouldn’t have a constant slope. Now, then, what does it mean for two lines to have a solution? That’s the point (my earlier defining of a point wasn’t just to be cute) where they intersect, and it’s necessary to know what a solution entails because this question is asking about how a solution would not exist. With all that in mind, we can start doing something with this problem, as opposed to what too many people, including this student I was helping, do: see it, mentally shrug, then move on.

Now, to go about finding k as the question asked, it would first be more convenient to compare these lines in the easier-on-the-eyes “y equals” form, where y is on its own side of the equation. Following some quick algebraic manipulation of the terms, we get:

y = 3x + 6 and

2y = -kx + 4 reduced to y = (-k/2)x + 2

Some more basic knowledge of linear equations helps you the rest of the way. The coefficient of the x term, that stuff in front of the x, is the slope of the line, also the rate of change of the line (flirting with calculus here). As we already said, linear lines have a constant rate of change because, well, they’re straight.

We know that first line has a slope of 3; it’s right there in front of that x. That second line has a slope of (-k/2) where k is an unknown constant, which just means it’s some fixed value. Add the definition of a constant to the list of things helpful to know for answering this question.

To find where there’s no solution to these linear equations—at last, the answer to the question—we set the slopes equal to each other, because identical slopes of different lines suggest they are parallel to each other, forever spaced equally apart, thus never intersecting, thus having no solution. Here’s some final algebraic manipulation:

-k/2 = 3

-k = 6

k = -6

Boom. Those linear equations have no solution when k equals -6. A big, bad, scary question is actually quite simple once you decipher the vernacular, identify the concepts at play, and understand those concepts sufficiently enough to work with them. In this case, knowing that the “solution” of two linear equations is the point at which they intersect and knowing that two lines will never intersect if they have the same slope were the keys to this problem. In other words, understanding math (Ding! Ding! Ding!) was the key to this problem, and that has nothing to do with how fast someone can multiply numbers in their head.

This question was merely an example; I picked it because it was among the harder questions of the SAT math section I just did in my latest tutoring session. In the same way I did in this article, I made that student realize his path to getting better at math is undergirded by conceptual awareness, not computational ability. Sure, cognitive firepower helps, but it’s far from the whole equation (unavoidable wordplay).

Problem-solving, in math as in everything, is a skill, an art even, but it’s easier to do with a bigger toolkit at your disposal. And you are completely in control of expanding and strengthening that toolkit. A person fluent in Mathmanese is better at doing math, the same way a person fluent in Arabic is better at writing it. So if you’re one of these people claiming to be forever challenged when it comes to math, you’re not. You probably just don't understand it. That’s not an insult; that’s a compliment of which to be optimistic about.

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