In Math as in Dance, Don’t Miss a Step, or Else You May Fall

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.

 

 

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4 Responses to In Math as in Dance, Don’t Miss a Step, or Else You May Fall

  1. Mariah says:

    I was reading this with interest (as I am currently teaching Finite Math) when my 7 year old daughter started reading out loud over my shoulder. Hearing your words from her lips made my mother’s day.

    When you dance, at first you are constantly counting in your head and paying attention to your feet. After a while, your feet start dancing naturally without you having to think about it and you focus on your posture, hands, partner, etc. As a professor, I hope that as my students advance, they are doing algebra “naturally”. However, I think it is essential that students should always be able to ask to be reminded what the steps are again.

  2. John Golden says:

    I definitely hear you. One of the ways I think about this is that I want to genuinely share my thinking (that’s modeling) rather than just tell you what to do. (Algorithm or worse when too specific.) And I ask my students to do the same, not just show their work but write out their thinking and explanations. Way more important to me for understanding their math than calculation steps. Sometimes ‘show your work’ is about obedience rather than understanding.

    I’d ask you to press you instructors; when they skip a step, stop them and ask them to share their thinking. How did they know to do that? How did it help?

    The other thought I have here is that I don’t really want my students doing my dance. I want to teach them to choreograph their own dances.

  3. Alison Gillespie says:

    Thanks for a great piece.

    This is a huge problem, and may explain why all over the US, math students who are not in the “advanced” math classes are failing at shockingly high numbers. Those who are advanced can make the leap. Those who don’t naturally take to math or need a bit more time to understand process fall on their butts, as you describe.

    Even more troublesome: many math classes in middle and high schools currently do not use textbooks, in either e-books or printed formats. It is unclear why this is — although in some districts it appears that it is due to the speed with which new math curricula are being imposed on states from the federal level. (I write that as someone who has no strong opinion one way or the other about Common Core, truly.) But anyway, a lot of places are now just have a teacher spew information verbally in class at break neck speed, and then sending students home with stacks of copied worksheets full of exercises to do for home. No texts to refer to at home, and nothing to help students review or try to grapple with tough lessons they might not have understood in the lecture explanation during class. No way to see all the steps from 1-8 so that you see where the leap was made during the lecture.

    I believe, based on my own experience, that what you describe happens a lot. I watch my kids struggle with math now and I recall the struggle myself. Combine that with the lack of textbooks and what are struggling students or even students that had a bad day to do when they get home and can’t for the life of themselves recall how a problem was done by the teacher? The answer sadly seems that most of the time they struggle, and then they either are lucky enough to be able to afford a tutor OR they just fail out of math.

    Very distressing.

  4. Maria Droujkova says:

    How can we invite students to be choreographers and composers, in addition to performers?

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