Treat people as if they were what they ought to be
and you help them become what they are capable of being.
–Goethe
You should really think of the student as innocent of understanding
until proven guilty by a preponderance of evidence
–Grant Wiggins
“Trig Has a Purpose!”
“Hey, Mr. Brock?!”
The sound of drills filled the workshop, and I looked up from where I was squatted, marking a two-by-twelve for a staircase. Katie waved where she and Alyssa were working, and I stood up and walked over to them.
“Yes?” I asked.
“We’re trying to secure this leg, and none of the two-by-fours that are left are long enough to run the length of the platform,” said Alyssa.
I looked at the pile of wood they had next to them on the floor, estimated the size of the longest pieces, and made a decision.
“So cross-brace it instead.” I replied, glancing at the height of the platform’s legs. “You both know how to do that.”
They shook their heads in unison.
“We tried that already,” replied Katie. “But Michael said this platform can’t have any bracing on the outside because of the facade we have to attach to it, and we can’t get the pieces of wood to lie flat enough when we hold them up on the inside to mark them to cut them.”
Alyssa nodded in agreement and reached down to pick up a short two-by-four.
“See?” She emphasized. “It’s impossible.” She held the length of lumber awkwardly against the leg and top of the platform.
“Okay, so we’ll have to figure out how to make it fit.” I told them. “Which one of you has a calculator?”
Katie said that she did and headed off to get it out of her book bag. Meanwhile, I pulled out my pencil and started making sketches on the plywood covering the platform.
“Should you be drawing on the set, Mr. Brock?” Alyssa asked, slightly concerned.
I turned and gave her the “you know better” look.
“Oh, right.” She said, coloring a little. “We have to paint everything before we’re finished.”
“Mmm, hmm.” I murmured and went back to drawing out my triangle. I finished putting in the angle symbols and then leaned back to study where the leg met the platform.
“Is this one of the ones they have to dance on?” I asked her.
Alyssa shrugged, but Katie answered “Yes” as she returned with her calculator.
Standing by the platform, they both studied me with puzzled expectation. We had worked together long enough that I knew what they wanted to ask; so I gave them an expectant look of my own.
“You tell me.” I told them.
“The braces need to be longer if they have to dance on it,” said Katie, not quite making it a question, “because that will make it sturdier.”
I nodded for her to continue.
“But we only have so much wood left.” She added.
“And if we want to have the budget to build the elevator into the stage floor and all the other stuff you’ve all dreamed up for this set…” I said deliberately.
“We have to know precisely how long to make the braces and where to attach them.” Alyssa finished, the understanding dawning fully. She looked at my drawing and asked, “So how do we do that?”
I pulled the tape measure off my belt, made two quick marks on the side of the leg and the middle of the platform, and determined the lengths from the corner to each mark. I then started writing numbers next to the lines on my triangle drawing and held out my hand toward Katie.
“Calculator, please.” I asked. Alyssa looked over my shoulder as I made additional marks on the wood and entered numbers into the small computer, and then she gave a little gasp.
“Oh my god,” she exclaimed. “You’re actually using the Pythagorean theorem!”
I paused, nonplussed, and just stared at her for a second.
“Yeeaahh…” I said slowly. “How else did you think we were going to figure out how long to make the braces?”
Alyssa ignored my sarcasm and shook her head, getting excited.
“No, no, Mr. Brock.” She said. “You don’t understand. I never knew math was actually good for anything before right now. They make us memorize all those stupid equations, and suddenly, maybe there’s a use for them after all.”
A look of understanding came over my own face, and I recalled my own first time that the abstract became real.
“Well, if you think that’s something, just wait.” I grinned, handing the calculator back to Katie. “Do you know how to make that thing do the inverse of a function?”
“Sure.” She nodded.
“Well, I don’t yet.” I told her truthfully. “So I’ll need your help. Cosine theta is 30 inches over 45 inches; what’s the angle, theta?”
She rapidly started to plug the numbers into her calculator, and I had to ask her to slow down. She looked at me, confused.
“How else am I going to learn how to do it?” I asked her.
She grinned and walked me through the calculations slowly as I looked over her shoulder for a change rather than the other way around, Alyssa just muttering over and over again in disbelief: “Trig has a purpose….”
“Forty-eight degrees, Mr. Brock.” Katie said finally.
“Okay,” I replied, turning back to the platform. “So now we know one angle to set the chop saw; now we need to figure out the other.” I started to reach for my pencil on the plywood, but in her enthusiasm, Alyssa grabbed it and was already busy adding and subtracting angle values.
“Forty two!” She all but shouted.
“But which direction on the saw?” I questioned, making sure both of them were thinking through the full 3-D nature of the problem.
“Opposite, Mr. Brock.” Katie chided lightly, reminding me that they had been doing this for a couple of years.
“All right.” I said, laughing at myself. “You both know what to do now.” I nodded in the direction of the rest of the set crew still putting the legs on their platforms. “When Allison, Suzanne, and Tory get ready to brace, teach them how to do it and pull anyone you need off from painting flats to help you if you need it. I’ve got to finish marking and cutting the stairs; otherwise it won’t matter what you all do because no one will have a way to get onto the platforms in the first place.”
I started to walk back to my own work when Alyssa said, “I could help you if you need it, Mr. Brock.”
I turned and shook my head.
“Only one ‘impossible’ per day.” I told her. “We’ll tackle stairs another time.”
Room for Other Minds
Historian Marshall Hodgson once wrote to the effect that all learners have an obligation to experience what they study as Other before they can ever claim any real knowledge of it. A Westerner studying Islam, he knew that he had to construct his metaphors of understanding in “terms available in his own mental resources”1 just like everyone does. But he recognized that neither he nor anyone else could claim that these metaphors accurately reflected the true nature of Islam–or any other subject–unless this mental construction process itself were done in such a way as not “to substitute [our] own… conventions for the original, but to broaden [our] own perspective so that it can make a place for the other.” Hodgson saw that not “until [we have] driven [our] own understanding to the point where [we have such] an immediate, human grasp of…a given position [that we] could feel ourselves doing the same” can any of us presume with any certainty to truly know whatever it is we are trying to learn. Thus, he knew that we must allow for the true différence of another to take hold in our minds if we ever want the discovery process to succeed, and he challenged his fellow scholars and other thinkers accordingly.
I share Hodgson’s argument here because I think it provides us with a critical insight about the educational process. We have already discussed at length that how teachers and students engage in meaning making is the key to educational success and that this “how” must involve immersion of one’s self in active engagement in a specific subject. But what Hodgson now allows us to see is that the study of anything can only lead to genuine knowledge of it when the learning process has caused the learner to become a new and different person, one who now incorporates what they have learned into a revised paradigm for making meaning about the world.
That is actually one of the dangers of a catechistic approach to teaching: that it equates memorization and recitation with learning and understanding. However, we need only look at a classic catechistic exercise such as reciting poetry to see why this equation is an invalid one. We all know that anyone can memorize and recite a passage of verse. In fact, doing so during the school years is practically a rite of passage. However, precisely because recitation is such an abiding ritual, each of us are also fully aware from our own personal experience that knowing the words to a poem and understanding what it means are two entirely different things, involving entirely different ways of interaction between the learner and the piece of verse. Everyone who has ever sat through an English class knows firsthand that unless you can identify with what you are reading at some personal level, it will remain just so many words on a page, and indeed reciting poetry is the exercise in educational torture we all remember it to be precisely because this activity is not how to enter into a sonnet’s world and experience its universe as if it were our own. A mere recitation remains as empty of meaning as a blank sheet of paper, and as long as a poem stays nothing more than an organized collection of words–just so many data bits for the brain to absorb and recall–then it will remain an unfathomed enigma and nothing more.
The only real way to understand a piece of verse is to learn the meaning behind its words, and to do that:
you have to open yourself to a poem…wait patiently upon it, and make an empty space for it in your mind [until] the work declares itself to you, steals deeply into the interstices of your being [and] line by line, note by note, phrase by phrase…becomes part of you forever.2
No one ever fully grasps a poem’s true meaning, its identity, until they have lived with it so intimately that the person can exhibit this meaning in his or her understanding of their own identity, their own sense of self. Hence, the critical difference between the mere recall of a poem and actual learning about it is that in the latter, a person’s newly created awareness of self now incorporates that poem as an integral part of that changed self.
Moreover, what holds true for poems holds true for everything else as well. Any successful student or a teacher of a foreign language will tell you that what defines fluency is the difference between the plu-perfect form of a verb being something someone has simply memorized for a test or to read a text–no matter how well–and something he, she, or they has made such an innate sense of his, her, or their own identity that he, she, or they can actually dream using it. As a matter of fact, those fluent in another tongue will even speak of their “Spanish self” or “Chinese self”–so totally does their understanding of the particular language affect their sense of who they are–and that is why total immersion in another language remains the only real way to master it.
Studying another language, of course, is not the only subject in which successful learning demands a new “self,” and I could just as easily have provided similar illustrations from every other academic field. But then–to state the obvious–that’s the whole point. The difference between the mere recall of information and genuine understanding of it is how it changes us. In order to genuinely learn, we must permit what we study to touch us and to alter us at our most basic level, otherwise how can we truly say we understand it? If what we learn doesn’t transform our actual attitudes and actions, it’s not learning, simply assimilation, and if newfound awareness doesn’t remold our understanding of ourselves and the world—does not truly change our paradigms of meaning in some fundamental way—it isn’t genuine knowledge, just data. Hence, like with a poem, how could anyone, for instance, fully comprehend the American Civil War until they’ve responded to the remnants of its racism in their own soul? Or how could they faithfully understand the forces of climate change and still leave lights on in empty rooms? Real learning fundamentally changes who we are and how we live in the world.
Light a Candle 