By A.K. Whitney, journalist. In 2009, Whitney went back to school to find out, once and for all, if journalists really are as bad at math as they fear they are; her blog about the experience, Mathochism, runs on Medium three days a week.
When you return to the classroom as an adult student, a big perk is that what seemed like an unreasonable demand back then from the instructor suddenly makes sense, because maturity means you’re better able to fit it into the bigger picture. For me, a longtime journalist who decided to retake high school math at a community college after decades of hating and fearing it, that demand was “show your work.” As a teen, I’d always sighed when the teacher marked me down for not showing how I’d worked out a problem on an exam or in the homework. Why was it necessary to take eight steps to show a triangle’s angles added up to 180? What a bore.
But 20 years later, going from pre-algebra to calculus, I finally understand why, and I credit dance.
Huh? Let me explain.
To get to the math building on my community college’s campus, I’d usually take a shortcut through the dance department. I’d walk down a long corridor lined with mirrored studios, and no matter what kind of music was blaring out the doors – salsa, tap, jazz – an instructor would always count out a beat before the students began.
“And a one, two, three, four, five, six, seven, eight!”
Hearing this every class day, I not only realized that numbers were everywhere, but also that learning how to solve a math problem was a lot like learning how to dance. In both, there’s choreography involved, going from step one to step two to step three. And, at least in the case of ballroom dances like the fox-trot or waltz or cha-cha, there is a strict order of operations.
You may be Please Excusing My Dancing Aunt Sally, not my Dear one (PEMDAS), but missing a step or doing it out of order will really mess up the end result. Or else, it will turn the dance into something completely different.
By thinking of math problems that way, I was better able to tolerate my instructors’ endless insistence that I show all my work, especially on tests. I finally appreciated that they needed to know I truly grasped the elements of the problem, and that I respected the strategy needed to solve it. True, I still find it tedious to prove in eight steps that a triangle’s angles add up to 180 degrees, but I now know it’s good practice for way more complicated proofs, where thoroughness is key. I also appreciate that precision is vital to math, and if eight steps is what it takes to be precise in a triangle proof, so be it.
That said, a major peeve of mine, especially as I got further from applied math and closer to pure, was when instructors, while solving a problem, would take a sudden leap. This might entail doing quick factoring in a polynomial, going from 6x +6 to 6(x + 1) without explaining why it was necessary, or assuming students had memorized an obscure trigonometric identity, then making the substitution in a long equation without mentioning it.
I realize these are very simple examples, but depending on where I was in my math education, to me this was the dance equivalent of doing a two-step, then suddenly getting spun and landing on my butt. It would always take me a moment to regroup, and by then, I’d been left behind, standing against the wall and watching as everyone else whirled by. At least in class, I could try and stop the instructor and ask him or her to explain. But I always felt guilty about this, since we never seemed to have enough time to really get into the material. That guilt was spurred by the fact that every professor I had, from pre-algebra on, complained about class time never being enough to really go into depth on anything, especially if students didn’t grasp the material right away. And yes, all of these instructors had office hours for those slower students, but I discovered those hours were just as chaotic as they were in class, only now students were cramped into a tiny office, craning their necks to see what the professor was writing in a notebook. But that’s another discussion for another day.
It was worse when such a leap happened in the solutions manual. For the record, I’ve never much cared for these manuals, preferring to puzzle things out on my own. But sometimes I would come across a problem I just couldn’t solve, where it was all a blur, and I couldn’t pick out one step from the next. Looking at the worked out solution was a way to slow things down and get a guide.
However, when that guide skipped a step without explanation, there was no lecture to interrupt, no office to stalk. I was usually able to fill in the blank, but the time I spent doing so always had a cost. Sometimes it was not being able to get to all the other problems I needed to practice before the next test, or, more important, it dented my still fragile math confidence, making me unsure when I had to perform. And that anxiety sometimes led to failure on exams because I couldn’t relax enough to solve harder problems without second-guessing myself. Then I would make silly arithmetical mistakes on the other problems because I was rushing to catch up.
Now, I know that my inner demons were never my instructor’s problem. But that didn’t stop me from asking a few math professionals I came across why they skip steps while taking students through a problem.
Not enough class time, one said.
Including every step gets very tedious, said another, and you can lose sight of the bigger picture.
It is the student’s job to fill in the blanks, and doing so is the best way to retain the material, said several, though at least one added that this method worked best in classes more advanced than calculus. It can really backfire before then.
And these are all excellent reasons. But they didn’t help that jarring feeling of being spun, of falling, of landing badly, that I experienced when I revisited math after 20 years.
I understand that in math, as in dance, you have to get up and dust yourself off. And I did. But far too many of us don’t, which is why so many of us give up. And giving up on math has far worse repercussions, not just individually, but for all of society, than not becoming proficient at the fox-trot. And unlike me, most self-proclaimed math haters never return to the classroom.
So I ask the instructors reading this to consider shifting their perspective as I did mine. I accepted how important it was not to skip steps, to respect the choreography, so that you could see that I understood what was going on. You may have the best of reasons, but when you skip steps without explaining why, people like me, unused to the elaborate choreography, will fall down. We’re still learning. Don’t assume we can see how you did that leap. And hopefully, we’ll soon be dancing as gracefully as you.