As I’ve been arguing during the year of this blog’s existence—and for some time before—we tend to think too narrowly about intelligence, and that narrow thinking has affected the way we judge each other, organize work, and define ability and achievement in school. We miss so much.
I hope that the portraits I offer over the next few months illustrate the majesty and surprise of intelligence, its varied manifestations, its subtlety and nuance. The play of mind around us. I hope that collectively the portraits help us think in a richer way about teaching, learning, achievement, and the purpose of education—a richer way than that found in our current national policy and political discourse about school.
Felipe, a student in a high school wood construction class, is the head of a team assigned to build a cabinet for his school’s main office. At this point in his education Felipe has built one small, structurally simple cabinet, and this current, second, cabinet has a number of features the earlier project didn’t. His storehouse of knowledge about cabinets, his “cabinet sense”, is just developing, and the limits of his knowledge reveal themselves at various points throughout assembly. Like this one.
Felipe is trying to record final figures for all the components of the cabinet—he and his co-workers are eager to begin assembly. He is working with Gloria and Jesus, and he is sketching with them one more three-dimensional representation of the cabinet, using several lists and a sketch he and the others had produced during planning.
When I approach the team, Felipe is looking back and forth from lists to the sketch and talking to his peers. He seems puzzled. He asks Gloria to get the first sketch they made of the cabinet. She retrieves it from her backpack and unfolds it. They study it for a moment. He says something to Jesus, then takes a tape and measures—as if to confirm—the length of the cabinet. Sixty-eight inches.
Felipe continues this way, double-checking, trying to verify, looking up occasionally to snag the teacher, Mr. Devries, who, however, is helping a group across the room. The source of this vexation is a discrepancy that emerged as Felipe, Jesus, and Gloria were listing final numbers: The length of the sheet of plywood for the bottom of the cabinet—this is found on the list of materials—is sixty-eight inches. But the length of the top panel—listed on another sheet—is sixty-seven inches. This makes no sense. As Felipe explains it, exhibiting a nice shift from numbers to their structural meaning, the top can’t be shorter than the bottom, or the cabinet will look like this: and here he makes an abbreviated triangle in the air with his hands. What’s going on?
Finally, Mr. Devries is free, Felipe goes to get him, and they confer. The sketch Felipe has is inadequate, is not detailed enough to reveal that the top panel rests inside notches cut into the top of the side panels. These are called rabbet cuts. Felipe’s discomfort resolves quickly into understanding. The bottom panel extends to the very ends of the side panels, but the top will be shorter by a half inch on each side, the dimensions of the rabbet cuts. Thus the mystery of the sixty-eight inch bottom and the sixty-seven-inch top.
The depictions of the cabinet in Felipe’s plans do not provide enough information—through graphics or numbers—to enable him to figure out the discrepancy in measurement between top and bottom. Yet he must rely on these lists and sketches, for he does not yet know enough about cabinets to enable him to solve the problem readily…or not to assume that the discrepancy is a problem in the first place.
Fast forward now to the next cabinet Felipe builds, a few months after the completion of the one we just witnessed. This time there is no confusion about the length of the top and the bottom panels; that earlier episode taught Felipe a lot. And there is evidence of his emerging “cabinet sense.” This new cabinet requires a plastic laminate over its surface. Felipe is laying the cabinet’s face frame over a long sheet of plastic and tracing the outline of the frame onto it. This will give him the covering for the frame but leave two fairly large door-sized squares of the plastic. Felipe stops, takes a step back, looks the cabinet over, and then reaches for his list of measurements and a tape measure. I ask him what he’s doing.
We’re short on laminate, he explains, and here you’ll have these two excess pieces of it cut away from the frame. We’ll need to use them. But, he realizes, they won’t cover the doors themselves, because the doors will be larger than the opening; they’ll attach onto and over the face frame. So, he’s trying to think ahead and picture where the as yet uncut surplus might go. What other, smaller pieces of the cabinet could be covered. That’s what he’s about to check. When I describe this event to the teacher, Mr. Devries, he smiles and says, “That’s how a cabinet-maker thinks.”
Felipe has taken algebra and is currently enrolled in college math; he knows what more advanced mathematics looks like. On the face of it, the math involved in cabinet assembly is pretty simple: reading a ruler; adding and subtracting (and, less frequently, multiplying and dividing) whole numbers, mixed numbers, and fractions; working with the basic properties of squares and rectangles. Yet, he says, "there’s so many pieces you need to take into consideration, otherwise, you’ll mess up somewhere.”
Felipe’s puzzlement, I think, is located in the intersection of traditional mathematics, learned most often in school, and the mathematics developed in the carpenter’s shop.
Traditional mathematics is in evidence throughout Mr. Devries’ workshop: from the calculations students do to determine cost per board foot to measurements scribbled on scraps of paper spread across the room. Considered from the perspective of school math—that is, if lifted from context and presented as problems in a textbook—the operations here would be, as Felipe observes, fairly rudimentary, grade-school arithmetic.
But as these measures and calculations play out in assembly—particularly an assembly that is unfamiliar—things get more complex, and thus Felipe and his crew move slowly and with some uncertainty. With an incomplete sense of a cabinet’s structure, Felipe must keep a number of variables in mind, arrayed in three-dimensional space, with each variable having consequences for the other. The top of the cabinet will be shorter in length by the sum of the two rabbet cuts in the side panels—but what about the width of the top? Will it rest in a cut in the back panel, and if so, what are the implications for the measurements of the back panel? Will the top extend into or onto the face frame? What does that mean for the face frame? And so on. In neurologist Frank Wilson's phrasing, this young carpenter is developing the ability to "spatialize" mathematics—and as Felipe notes, that means taking "so many pieces…into consideration." Mr. Devries tells me that he has students taking calculus who have a hard time with such tasks.
There is a small but growing research literature on mathematics in the workplace—from the tailor’s shop to the design studio—and a few of these studies focus on carpentry. Listening to Felipe puzzle over the nature of the mathematics of assembly led me to look more closely at the math in Mr. Devries’ shop, and what I saw matched earlier studies, some of which were conducted in other cultures, such as in South Africa, suggesting some cognitive commonality to the way carpenters do the work they do.
One of the findings of this research is that a wide range of mathematical concepts and operations are embodied in carpentry’s artifacts and routines, and in ways suited to the properties of materials and the demands of production. The carpenter’s math is tangible and efficient.
Take, for example, measurement. The ruler and framing square provide measurements, but so do objects created in the shop: one piece of wood, precisely cut, can function as the measure for another. Tools are also used as measuring devices. A sixteen-inch claw hammer laid sideways on a wall provides a quick measure for the location of studs in a wall frame. And carpenters use their hands and fingers to measure and compare. (“I use my forefinger and thumb for calipers,” reports master woodworker Sam Maloof.) They develop an eye for length and dimension and for relations and correspondences.
Working in the shop, the young carpenter learns a range of other mathematical concepts: symmetry, proportion, congruence, the properties of angles. Planing straight the edge of a board, cutting angles on the miter box, laying out the pieces of a cabinet’s face frame to check for an even fit—through these activities, Mr. Devries’ students see mathematical ideas manifested, and feel them, too, gaining a sense of trueness and error. Fractions were never more real to Felipe than during the episode with that cabinet top.