About the Blog

I will post a new entry every few weeks. Some will be new writing and some will be past work that has relevance today. The writing will deal in some way with the themes that have been part of my teaching and writing life for decades:

•teaching and learning;
•educational opportunity;
•the importance of public education in a democracy;
•definitions of intelligence and the many manifestations of intelligence in school, work, and everyday life; and
•the creation of a robust and humane philosophy of education.

If I had to sum up the philosophical thread that runs through my work, it would be this: A deep belief in the ability of the common person, a commitment to educational, occupational, and cultural opportunity to develop that ability, and an affirmation of public institutions and the public sphere as vehicles for nurturing and expressing that ability.

My hope is that this blog will foster an online community that brings people together to continue the discussion.

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Thursday, March 5, 2009

Portraits of Thinking: A Novice Cabinetmaker


Here is a second story about cognition in action, a glimpse at a young man developing skills as a cabinetmaker. For those of you who missed the previous entry where I discuss the purpose of these portraits of thinking, I’ll repeat two introductory paragraphs now. If you did read the earlier entry, you can skip right to the story of Felipe, which is drawn from The Mind at Work.

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.”

* * *

Several times during the construction of the wall cabinet with the puzzling sixty-seven inch top, Felipe would comment on the mathematics involved in cabinet assembly. And I asked him about it myself. His comments were a bit contradictory, and the contradiction resonated with something that was intriguing me as well. At times he would note that the math is “simple,” “just numbers,” “only fractions.” At other times, though, even within the same few sentences, his face registering perplexity, he would observe that “a lot of math is involved” and that “it’s difficult.”

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.

11 comments:

Tamsen said...

I completely agree on the point that the academic world looks too narrowly at intelligence. What do we do with students who do not see (or may not want to see) the beauty in Anna Karenina or the intricacies of a chemistry equation, but they are complete geniuses when it comes to things like mechanics and cabinetry? Should we continue to cram the literature cannon down their throats when they are not inclined to enjoy or understand it?

Robert Bradley said...

I am interested in Felipe’s contradictory thoughts on what he considers “simple” but also “difficult” mathematics. It is clear that the isolated mathematic equations are simple, but the combination of many mathematical equations creates some difficulty. This reminds me that understanding concepts of a particular subject is only a step in the process of learning. For the learning process to continue, there must be continual steps to put concepts into practice.

ellzabethj said...

Felipe’s story is a perfect example of how subjects can be made “tangible.” It is my sense that all subjects – not just mathematics - should be made as tangible as possible. When students are able to work with and apply concepts in a hands-on way, the relationship to the subject matter becomes more personal and concrete. Knowledge becomes a tool to be used - not just a list of abstract facts or procedures to memorize, but a tool to achieve a “real” end product.

Suzanne said...

I really enjoyed reading this post since I have been thinking about intelligence in this way for a long time. I have been teaching English to college students for almost twenty years and in that time I taught myself how to crochet, knit complicated garments, and spin my own yarn. The intelligence I used to learn these skills was different than the intelligence used in my academic pursuits, yet my thrill at accomplishment was even more satisfying in many ways. Learning these skills while at the same time my students were learning the skills of writing, reading, and critical thinking that I was trying to teach them helped me to identify with their process of learning.

Our society definitely undervalues this type of learning and neglects to engage students in practical pursuits of creation that could help them develop critical thinking abilities in ways we never imagine when we label or segregate these knowledge areas.

I find it interesting that many of the knitting and spinning blogs I read are written by college students and graduate school students who must be craving a way to express their intelligence in more physical and practical ways. Some of the writing they do on these blogs is incredibly academic and analytical, yet they are writing about something real and creative. We need to find a way to bring some of this great learning into our classrooms.

Megan said...

I love to see experiences such as Felipe's. He sees the contradiction of simple but difficult math, but he is working through it with real, kinesthetic assignments. With the budget cuts of California, I am afraid that these experiences will also be cut. How can teachers provide these assignments while managing standards, curriculum, and overcrowded classrooms? Teachers will have a hard time, but what happens to the students like Felipe who might miss out on such influential discoveries?

Bobbi Elaine Faulkner said...

I am an ESL/English/Reading teacher pursuing a doctorate in Reading. Therefore, a lot of your stories really resonate with me because they represent "my" kids. I, too, feel that there are so many in-roads to a person's mind and intelligence, but academia views intelligence so narrowly. I am heartened, however, by the development of career center high schools. At the same time, these schools make me feel ambivalent because students learn literature, math, history, and science revolving around their chosen trades. On its face, that's a wonderful propsition--learning in context; gaining applicable content-area knowledge. My qualm is that these students will miss out on a rich body of "classic" knowledge. By "classic," I don't mean the cannon per se, but just reading for pleasure's sake, learning about invigorating scientific inquiry, etc. In some ways, I feel that we just assume that these kids aren't capable/interested in that type of education. (Reminding me of your vocational ed. experiences discussed in Lives on the Boundary.) On the other hand, how is this different from Germany's educational system, where students decide early-on their interests and future pursuits and begin training for them?(and that's not a critical observation; simply a comparison with a system that seems to be functioning well). What's your take on this issue? What do you know about these schools/programs? How do we strike the proper balance? As always, there's so much call for reform these days, a push to 21st Century learning, though many of us don't yet grasp all that this new technological age entails. I hope to be a leader in education one day, so I thought I'd tap your mind, as you are a wonderful leader in the field today!

Emily King said...

It is wonderful to see Felipe taking a kinesthetic process in learning difficult aspects of mathematics. By hands-on trial and error, he is learning through his mistakes. I believe that it is only through actually doing, practicing and experimenting with new concepts or ideas (and we might and must fail at first) that we truly learn—gain knowledge that will genuinely stay with us. This can be seen through Felipe’s confusion with the first project and ease with the second (aside from the new problem).

The fact that this story was about a wood working student learning math through carpentry really hit home for me. My father excels in carpentry work and has made a living through designing and building various projects over the years. And in my mind, through witnessing his process of thinking and executing challenges, my father is one of the most brilliant men I know. I am forever amazed at his knowledge of mathematic concepts far beyond my learning and his ability to picture 3-D objects drawn on paper. His dimensional pictures are precise and easily become real when constructed in our garage. I think it is important to remember that skills such as this, although it may not be seen as “book smart,” indicate a very intelligent human being. One that is able to brainstorm, organize, contemplate, problem solve, and complete an objective. It takes a particular type of individual with an aptitude for a specific type of knowledge and learning to be a carpenter. And I am afraid that our education system does not support these types of students as well as it should. How can we, as educators, better assist kinesthetic learners? In what ways can we take a more hands on approach to learning in order to accommodate all types of intelligences? And then the question becomes, who decides what is considered intelligent?

Anonymous said...

This story of Felipe is an interesting reminder that there is sometimes a difference between book learning and putting that learning to practice out in "the real world." Felipe's experience shows how learning doesn't stop with reading and classroom assignments, but continues with exposure and interaction with projects like the one he embarked on.

Anonymous said...

Thank you for the post!

Although it was not necessarily the main argument of your post, what I most enjoyed was seeing Vygotsky’s Zone of Proximal Development in action. Your story explained that Felipe and his peers struggled to understand the measurements of the cabinet that they were building for his school’s main office. You claim that they had not yet developed the “cabinet sense.” Therefore, the combination of the inadequate cabinet depictions and the students’ actual development levels were not enough to complete the task at hand. However, with the guidance of their teacher, Mr. Devries, the students were able to cognitively understand the measurements and later complete the project. Vygotsky shows us that students can rise above their actual development level with the guidance of a teacher or more advanced peers, and your story demonstrates this perfectly. With a little help, Felipe’s “cabinet sense” was enhanced, and he was able to successfully apply his newly learned skills to the next project that came his way. How exciting!

--EMO

Jessica Hadid said...

I have come back to re-read this post numerous times, drawn both by the topic of woodworking and the topic of how our minds adjust via mode according to situations. As someone who has struggled with mathematics but managed to design and build many things with wood, I can attest to the notion that “spacializing” math in the woodshop is altogether different from tackling mean numbers on the page. For some it may be the long needed chance to succeed. There is something I’d like to add: making things, creating from raw material an object of personal value, is a kind of revival of self. There is discovery in such creation, a discovery that one is capable in ways of which he was unaware; a sense of satisfaction in the accomplishment. As well, in creation there is purpose—which is everything in the world of education.

Anonymous said...

I agree with both Tamsen and Robert Bradley. There is a world of difference from having academic knowledge to worldly knowledge. An example of this comes from my own experience. I am terrible at math, I have always just squeaked by with barely passing grades. When I was in high school I would work with my uncle laying tile. He taught me how to measure the floor and how to calculate how many boxes of tiles I would need to buy. I mean this is stuff I was learning at the same time in high school, but when it came down to it I would fail a test in school, but correctly figure out how many boxes of tile I needed for the floor. I think that academics is more abstract, so some people struggle with it a little more and when it comes down to worldly knowledge its concrete, things people can actually get their hands on and actually feel out the task at hand.