Math with Bad Drawings

The Three Barriers to Deep Thinking in School


Almost a decade into my teaching career, I’ve learned a lot—about recurring decimals, British slang, the life cycle of fidget spinners. But one lesson I seem to relearn in new ways every day: Deep thinking is a very, very delicate flower.

It blooms only under rare and perfect conditions, when you’ve given the seedling absolutely everything it needs.

There’s no perfect recipe. What gets my 6th– and 8th-graders’ thoughts blooming might flop with my 7th-graders. This work is wonderfully and maddeningly specific. Each seedling presents its own unique and irreducible case. The best you can do is kneel down in the soil and try to help it along.

Even so, I find a few recurring themes: three crude reasons why deep thinking fails to bloom, and the hardy but colorless perennial of “rote learning” surfaces instead.

  1. As students, we seek the cognitively easier path.

Earlier this year, I had a typical conversation. A 7th-grader wasn’t sure why 3√2 + 4√2 should equal 7√2.

“Well,” I said, “what’s 3x + 4x?”



“Because you add the 3 and the 4, and keep the x the same.”

“That’s an accurate description of what you’re doing,” I said, “but let’s try to figure out why it’s true. What do ‘3x’ and ‘4x’ mean?”

They mean, of course, “three groups of x” and “four groups of x.” That totals seven groups of x, no matter how large or small x happens to be. That’s elemental, but not elementary. It demands that you (1) think about specific cases; (2) look past their superficial differences to the underlying similarity; (3) articulate a general principle; and (4) translate your discovery into algebraic notation.

Or… you can ignore all that, and just learn a rule for moving symbols around.

Not to scale. In reality, there are about 6000 times more irrelevant details.

“Oh, I get it!” he said before long. “You just add the 3 and the 4, and leave the √2 the same.”


  1. As a teacher, I seek the administratively easier path.

Back to my 7√2 student: why had I, his teacher, put him in such a pickle to begin with? Why was I asking this student to extend a rule that he didn’t even understand?

Well… because that’s what came next in the syllabus.

Sure, I could have found a better personal activity for him. Or I could have found a richer task for the whole class, so that he could explore this idea while his classmates explored others. Or I could have stayed with him—collating numerical examples in a table, producing visual and verbal models, testing his symbol-based rule on cases where it would fail—until he developed an actual understanding of why 3x + 4x = 7x.

So why didn’t I?

Because lessons are short, teaching is hard, and I had a classroom full of 7th-graders to manage.

Symbol-pushing isn’t just easier for students. It’s easier for me. It takes less planning before class, less improvisation during class, and less mop-up with struggling students afterwards.

To help a room of students think deeply—that’s no easy task. To help them learn superficial facts and mechanical rules? Well, that’s a heck of a lot easier.

  1. As assessors, we seek clear-cut standards by which to rank students.

It was there in the room, as I spoke with my 7√2 student. I’m not sure I can pinpoint where—hovering by the ceiling? lurking under my desk?—but it was there:

The system’s need to assess.

Schools play a lot of roles in society, and one of them is the crude business of sorting. In June, the school’s students all take a year-group test. Top scorers win prizes and are called on-stage in front of the whole school. Low scorers feel the gut-punch of failure, no matter how meaningful or meaningless the test is.

Three years after that, the same students sit the IGCSE exams. Top scorers have a better shot at prestigious university degrees. Low scorers have a worse one.

This stuff matters.

The tests are written to be “objective” and “fair,” which means they ask for scripted performances of technical skills rather than for flexible improvisation. On such tests, deep thinking can be more an impediment than an aid.

So what is there to be done? How do you help healthy flowers grow in a climate that can feel so ill-suited to them?

Well, that’s called teaching, and I’m still learning how to do it.

But I’ve got some ideas.

To overcome the first obstacle—students’ preference for easier thinking—I’ve got to give them motive and opportunity. I’ve got to help them see the steeper path as an exciting adventure, not a pointless side quest.

To overcome the second obstacle—my own preference for easier administration—I’ve got to play it smart and conserve my energy. I’ve got to avoid the mire of self-created busywork, and lay the groundwork (with routines and class culture) to make open-ended tasks go smoothly.

And to overcome the third obstacle—the system’s preference for clear-cut assessment—I’ve got to fight a multi-front war. I’ve got to seek better, richer, more varied assessments. I’ve got to help students see their results not as irreversible judgments but as guiding feedback. And—hardest of all, in a system that puts all kinds of pressures on teachers and students alike—I’ve got to know when to hold ‘em, and know when to fold ‘em.

Now, the goal isn’t for students to think deep thoughts every minute of every day. That’s as unsustainable as a full night of top-intensity dancing. You need cool-down songs, slow dances, chances to catch your breath.

But I know this much: I want my students dancing as hard as they can.