Paper Skills & Puzzling Skills

Math is cumulative by nature. And what do you think accumulates most often by the time a student hits middle school?

Learning gaps.

If there’s one age group I get referred to the most in my work as a private tutor, it’s students in 6th and 7th grade whose parents are befuddled by the sudden struggles their kids are facing. After all, these learners did just fine in the math they saw at the elementary level. Why so much stress and angst all of a sudden?

There’s a host of reasons, of course — hormones being one of them and changing schools another — but big emotions and new curriculums aren’t the only factors that make middle school math persnickety. It’s also the onset of algebra that can leave students spinning.

I don’t know about you, but my negative opinions about my math-worthiness cropped up yon about 6th grade.

Before then, any awareness I had of my mathematical abilities came straight from my standardized test scores. The number next to “Mathematical Proficiency” was lower than the one next to “Language Arts,” so — in my young mind — that meant reading was my thing and math wasn’t. For a while, I was okay with this. My grades in math weren’t in the basement, and I understood enough of the lessons for the subject to feel like a second language, not a completely foreign one — but then I hit middle school. That’s when I went from thinking I was just better at reading to deciding that I was downright bad at math.

Of course, I was wrong about this — but try convincing a twelve-year-old of that.

Here’s what happened to me, and what I see happening to so many students:

My elementary school math life was all about arithmetic. I learned steps. Memorized them. Rinsed and repeated. On a regular basis, I used what I’ve come to think of as my paper skills — ones that allowed me to accomplish computations like long division and multiplication without a calculator. By definition, these skills required a healthy degree of executive functioning: the ability to organize, tap into my working memory, sustain focus, and successfully memorize.

I was good at that stuff, so I leaned on those skills and thought they were all that math was made of.

I was wrong. As it turns out, math doesn’t just require paper skills. It requires puzzling skills, too.

What are puzzling skills? They’re the ones we use to reason abstractly and to logically deduce. They’re different from the skills we use to complete paper and pencil arithmetic.

Here’s what I mean:

A problem like this one requires paper skills: 573 x 654

A problem like this one calls for puzzling: Order the values: -0.18; 1/8; 1.8, -1/8

This guy demands both: -5(x +6) + 4x = -9x + 2

Do you see the difference?

Puzzling skills were the ones that didn’t come naturally to me early on, and so I didn’t develop them, and when pre-algebra arrived, I wasn’t prepared to reason — I was just a proficient human calculator. I needed two sets of skills but I only had one — much like the equations I repeatedly solved incorrectly, I was out of balance.

I wish I’d understood then that I wasn’t terrible at math. I was just working with one set of tools when I needed two.

I see this pattern play out with my students over and over again.

Either the puzzling skills come naturally to them or the paper skills do, but whichever one feels more comfortable, that’s what they use early on — and far, far too often — while the other set of skills, the ones they actually need to strengthen, are left in the dust.

This imbalance is why students can make it through elementary math, but then hit an absolute wall round about the onset of pre-algebra. That’s when computations regularly become too complex to compute mentally and too abstract to rely on any given set of steps. If a student hasn’t worked on his paper skills, he’s going to struggle with seeing multi-step computations through, either because of organizational issues (which cause precision mistakes) or because there are essential paper steps he hasn’t learned along the way. Conversely, if a student has always followed the steps given to complete her pencil and paper computations but never puzzled through why those computations work and what happens to values when we operate (hand up!), she’s going to have a very difficult time with things like balancing equations because there’s no one-size-fits-all set of steps to follow.

So, what’s a student to do?

As was the case with asking the questions what, why, how, and when, no one set of skills is more important than the other. We need both paper skills and puzzling skills to help us get through. If we have been overcompensating with one or the other, eventually things will come to a head. To combat this problem, we would all do well to investigate when we are using each set of skills, whether a problem or lesson demands one or both, and challenge ourselves to discover where our strengths and struggles nest.

Here’s an encouraging note to end on.

One of my students is in 11th grade these days, but we started working together when he was in middle school. He’d always struggled with neatness and focus but it was obvious he had a real head for numbers. C’s and D’s were all he expected of himself in math back then, and his confidence was in the basement. Frustrated and angry, he thought I was nuts when I told him he was actually quite good at math — it was just his skills on paper and his patience that needed polishing. To his credit, he worked hard and accepted advice. He realized how good he was at puzzling, and he put real energy into improving what was happening on his paper. All these years later, his work is organized and math has become his favorite subject. Frankly, he’s a different kid. He laughs when we work, he gets excited about the concepts he’s learning, and he feels genuinely hopeful about his future in math.

Somewhere along the way, he replaced insecurity with self-awareness. He identified his strengths and used them to balance out his struggles. In doing so, he cultivated a healthy dose of self-confidence.

What a little miracle, don’t you think?