When you were a youngster, you almost certainly learned a little about numbers and counting before you got into school: 1, 2, 3, 4, 5, . . . and so forth. This was the first rung on the crooked ladder —the first of your steps toward learning more mathematics.

And it was just about everyone’s. No doubt, while there can be—and are—significant differences in students’ mathematical background knowledge at the age of 5 or 6, virtually everyone that you know or have known or will know started in or will have started in the same place in math: with the positive whole numbers and the operation of counting discrete quantities.

The next few rungs of the ladder we also mostly have in common. There’s comparing positive whole numbers, adding and subtracting with positive whole numbers, whole-number place value, some geometric shapes, and some measurement ideas, like time and length and money. And to the extent that discussions about shapes and measurement involve values, those values are positive whole numbers.

Think about how much time we spend with just discrete whole-number mathematics at the beginning of our lives—at the base of our ladder, the place where it connects with the ground, holding the rest in place. This is not just us working with a specific set of numbers. We learned, and students are learning and will learn how math works here, what the landscape is like, what operations do. This part of the ladder is the one that holds up students’ mathematical skeletons—and it is very much still a part of yours.

I would like you to consider for a moment—and hopefully longer than a moment—the possibility that it is this beginning, this crooked part of the ladder, that is primarily responsible for widespread difficulties with mathematics, for adults and children. I can’t prove this, of course. And I have no research studies to show you. But I’ll try to list below some things that reinforce my confidence in this diagnosis.

And Then We Get to . . .

For starters, there are some very predictable topics that large numbers of students often have major difficulties with when they get to them: operations with negative numbers, fractions, and division—to name just the few I have heard the most about. Well, of course students (and adults) have trouble with these concepts. None of these even exist in the discrete positive whole-number landscape we get so used to.

Ah, we say, but that’s when we extend the landscape to include these numbers! No, we don’t. We put the new numbers in, but we make those numbers work the same way as in the old landscape—we put more weight on top of the crooked ladder (I’m challenging myself now to mix together as many metaphors as I can). So, multiplication just becomes addition on steroids—super-charged turbo skip-counting of discrete whole number values; division cuts discrete whole-number values into discrete whole-number chunks with whole-number remainders, more skip counting with negative numbers, and fractions are Franken-values whose meaning is dissected into two whole numbers that we count off separately.

“But we teach our students to understand math rather than follow rote—” No, we don’t. I mean, we do. We think this is what we are doing because the crooked ladder is baked into our mathematical DNAs (3! 3 metaphors!). So, we say things like, “I’m not going to teach my students the rules for multiplying and dividing fractions! No invert-and-multiply here, nosiree! I’m going to help them understand why the rules work!” Then what do we do? We map fraction division right on to whole-number counting: how many part-things are in the total-thing? And we call it understanding.

Don’t get me wrong. Teaching for understanding is much better than teaching procedures alone. My point is that most of the metaphors we are compelled to draw on (and the ones students draw on in the absence of instruction) to make this ‘understanding’ work—those involving concrete, discrete whole-number “things”—are brittle. And though they might be valuable, they certainly don’t represent “extending the landscape” in any appreciable way that opens up access to higher-level mathematics. Our very perception of the problem of ‘understanding’ can be flawed because we are developing theories from atop a crooked ladder.

(It’s right about here that I start hearing angry voices in my head, wondering what we’re supposed to do, “bring all those advanced topics down into K–2? Huh?” And this is just what a crooked ladder person would wonder, since he has no experience with any other ladder, and no one else he knows does either. The only possibility he could fathom is to take the rungs from the top and put them on the bottom.)

And About Those Theories . . .

Anyway, secondly, theories. You may have noticed that there are a lot of folk-theories and not-so-folksy theories trying to explain why students and adults seem to have an extra special place in their hearts for sucking at math.

The theory I hear or see the most often—the one that, ironically, doesn’t believe it is ever heard by anyone even though it is practically the only message in town—is that mathematics teaching is too rote, too focused on rules and procedures, obsessed with “telling” kids what’s what instead of giving kids agency and empowerment and self-actualization and letting them, with guidance, discover and remember how mathematics works themselves. It’s too focused on memorization and speed and not enough on deliberate, slow, thoughtful, actual learning. Et cetera.

I guess that seems like a bunch of different theories, but they really come most of the time packaged together, like a political platform. And they’re all perfectly serviceable mini-theories. I think they’re all true as explanations for why students don’t get into math. But they’re also true in the same way as “You’re sick because you have a cold” is true—tautologically and unhelpfully.

Students and their teachers eventually fall back on the rote and procedural because after a certain point up the crooked ladder, trying to make discrete whole-number chunky counting mathematics work in a continuous, real-number fluid measurement landscape becomes tiresome and inefficient. A few—very few—manage to jump over to a straighter path in the middle of all of this, but a lot of students (and teachers) just kind of check out. They’ll move the pieces around mindlessly, but they’re not going to invest themselves (ever again) in a game they don’t understand and almost always lose. In between these two groups is a group of students who have the resources to compensate for the structural deficiences of their mathematical ladders. Some of these manage to straighten out their paths when they get into college, but for most, compensation (with some rules and rote and some moments of understanding) becomes the way they “do math” for the rest of their lives. These latter two groups will cling to procedures for very different reasons—either because screw it, this doesn’t make any sense, or because whatever, I’ll get this down for the test and maybe I’ll understand it later.

And Their Solutions . . .

The remedy for the inevitable consequences of ascending a crooked ladder—again, the one I hear or see the most often anyway—is to kind of take the adults and the “telling” out of the equation. And to make sure the “understanding” gets back in. And again this is more like a platform of mini-proposals than it is one giant proposed solution. And they work just like “get some rest” works to cure your cold—by creating an environment that allows the actual remedy to be effective.

So leaving students alone is going to be effective to the extent that it does not force students to start up a crooked ladder. But a lot of very different alternatives are going to be effective too. Since the main problem is a kind of sense-making exhaustion inside a landscape that makes no sense, any protocol that helps with the climb is going to work or have no effect. So, higher socioeconomic status, higher expectations, increased early learning, more instructional time, more student engagement and motivation—they’re all going to work to the extent that they sustain momentum. The real problem, that it shouldn’t require that much energy to move up the ladder in the first place, remains.

But Don’t Ask Me for a Solution

I don’t have any concrete things to propose as solutions to the crooked ladder problem, and if I did I wouldn’t write them down anyway. We can’t do anything about our problem until we admit we have one. And while we are all willing to admit of many problems in K–8 education, I don’t think we’re admitting the big one—the ladder at the bottom is crooked. The content, not the processes, needs to change.

Image credits: tanakawho, Jimmie, CileSuns92.