Crystal Lessons

by Michael Rossman


        It's the first day of school, and I greet my new students with crystals for beauty and wonder. Massive clusters of amethyst, dense brassy pyrite cubes, miniature cityscapes of transparent quartz spires, fanged clusters of dogtooth calcite, creek-cobbles split open to books of shiny mica, fluffy asbestos, raw garnet crystals as big as my thumb, tiny garnets in eclogite matrix from down deep where the diamonds grow, and on and on -- a treasury salvaged from yard-sales, the flea-market, and rock-shop leavings for a buck or three apiece, plus some modest finds of my own afield.

        The table grunts as I set the box down to start laying out my specimens. It's heavy enough to anchor a curriculum, as I intend. For I offer the crystals' beauty as a blessing on our enterprise together, not simply today but for the next four years. This beauty is a hook, to draw my students toward inquiry and power -- or, as educators say, to motivate them to learn about the molecular basis of matter -- a subject we shall play with throughout our time together, illuminating the mysteries of life as we explore them.

        It is deviant to emphasize "chemistry" with second-graders, years before children are thought ready for its abstractions. Still I jump on the subject from the start, not in blind regard for its fundamental importance, but in practical opportunism -- because so many of its basic ideas can be expressed so clearly, concretely, and dramatically through simple demonstrations and experiments, and can be connected so directly to basic processes in physics and biology. This is solid meat for the mind, and we shall chew it intensively during our first four weekly classes.

        I have worked my way to this introduction by feel over many years, by finding what fit together and made sense in my way of teaching, in schools that left me free to explore. I am somewhat surprised myself, at how austere and demanding a welcome it seems, to tender spirits hardly yet able to consider an idea seriously. For years I used an easier, soft-edged hello -- devoting many of our early sessions to the tender introduction and handling of local fauna, from sowbug to snake to baby possum -- not only to familiarize them and begin systematic exploration of the animal order, but more deeply to engage my students individually, and engage us all in a community of caring. Yet though I still pursue this vital curriculum throughout our years together, I have abandoned it without regret in favor of a crystal introduction, which serves as well to bond us in the passion and adventure of inquiry together.

        Since my main purpose here is to present a later, more complex lesson, I will only sketch this second-grade introduction, emphasizing the process of making molecular behavior concrete in their imaginations. So this next section is rather dry and skeletal; but what follows it is juicier.


[A Crystal Introduction]

        Today my second-graders will revel in the wealth displayed here. They will fondle the crystals' strong colors and definite geometries, begin to sort out which are like and unlike which others in what ways, and learn some names of shapes and species to express this. At the end, I pose our project: Would you like to grow crystals yourselves? Of course they would, I've got them hooked, I'm the magician with the keys to the kingdom. I promise them success, crystal magic of their own, and warn them what it will cost: "But you've got to understand how crystals grow, to be able to grow them well. Do you want to learn this?" Who could resist?

        I begin next time by handing round fine crystals of blue vitriol, cheap common copper sulphate -- vibrant blue gems, their precise rhomboidal faces exciting to the eye. "These are the crystals we'll grow. Now watch." I smash one crystal into pale blue powder with a hammer, despite their cries of distress, and drop a large pinch into a small glass of water and stir. The powder turns gem blue as it settles on the bottom, and then slowly disappears, its tint diluted in the water above. "What happened?" We take a good while to conclude that the blue stuff is still in the glass but is no longer together in a lump. I put drops on some slides and dry them on a hotplate with slow care. What remains is whiter than the powder I started with, but a drop of pure water turns it gem blue again, proving our conclusion.

        I slip the slides onto the stages of the waiting microscopes, focus, demonstrate how to look into their tubes, and invite my students to observe. Midway in the dried puddles, the tiny rhomboidal outlines of a few microscopic crystals stand out clearly. A mystery of growth has happened before our eyes; can we learn to control it? We think again about how the blue powder dissolved, and conclude -- with my help -- that it was made up of bits too small to see, that now hang somehow suspended in the water, which somehow is made of tiny bits too. I offer the name "molecules" for such bits. Almost everything we can see is made of molecules, so many different kinds. We will study only a few simple kinds, to learn how molecules behave. But since we can't see them even with microscopes, we'll need other tools to look at them with our minds and imaginations, before growing crystals ourselves.

        For starters, I offer a simplified model of the very crystal we shall grow – a 4x3 array of green balls joined by wires into a rhomboid, with two more such layers stacked slantwise above. This is almost the least structure necessary to concretize the vision of a crystal as being made of molecules. I take care to explain that it is just an aid to our imagination – for molecules aren't wooden balls or even round, and what joins them in solids isn't wire. I pass round small magnets and set my students to feeling how two pull on each other, to get a sense of the invisible forces symbolized by the wires. Collecting the magnets, I lead them to consider the model's meaning: a crystal is molecules arranged in regular array, the self-sought nature of solids.

        To our next session I bring ice-cubes, hotplate, and chilled beakers. I drop a cube on the hotplate. We watch as the solid water gets hotter and melts to a liquid that gets hotter and evaporates to invisible water vapor, that condenses back to visible liquid as I catch it by cooling on a chilled beaker, and get them to agree that it would freeze back to the solid if we made it even colder. Thus we encounter the traditional Three States of Matter, and the processes and manners of their metamorphosis into each other. I draw the austere mandala on the whiteboard in vibrant colors -- seven words, four curved arrows -- and we go through its integral circuit of transformation several times to make sure we grasp the terms and their meanings. After letting the kids play with ice-slivers on the hotplate and try to condense vapor from the steamy air themselves, I take them through the mandala again -- this time with iron, imagining a bar melting in a furnace and heated even to a vapor that can be condensed and frozen to solid iron again -- until they can chant its terms confidently.

        "Now let's think about molecules." Later, while we're growing our crystals, I'll evoke the ancient Greeks who had the novel  ideas that stuff might be made of bits of stuff too small to see, and that its many varieties might be formed from a basic few kinds of bits called elements.  I'll name a few elements, and put brightly-colored balls together to model our copper sulphate molecule. But for now I must stay concrete, so I wiggle my fingers to demonstrate the basics: What heat is, is molecules moving. “Hotter” means they’re moving faster, “colder” means moving more slowly.

        My fingers are water molecules. When they're cold they just wiggle gently, because molecules are always moving; but they stay together in formation as solid ice. My interlaced fingers begin to quiver faster. As they warm further, they loosen completely, and my hands begin to roam over each other like groping octopi, the molecules dancing around each other with more abandon as I chant hotter, hotter, until my fingers evaporate to fly in quick flickering motions through the air. Quick, turn them back to liquid water, what's the magic word? “Condense,” they say, and I chant colder, colder, liquid now as the octopi reform and slide around each other again, colder colder freezing solid as they settle into natural formation, still quivering. "Got it? Now let's do it together." I lead them through it twice, in merry, careful play; they learn the hand-dance quickly. "Good. Now let's go to the courtyard."

        In the yard, we invest our repressed animal energy in the Dance of Changes of State. The class is large enough to make a sufficient mob as I define the rules of molecular civilization: Keep your motions under control. We join together to make a solid, by each hand of each child finding two different others to grasp in triangular bond, with as few hands as possible settling for less. Though cold, we still sway and feel miniscule quivers, but our union is secure. As we warm we wiggle more rapidly, our grips loosening as we melt to water, hands retaining gentle sliding contact with each other's arms and bodies as our feet grow energetic and take us sidling randomly around each other in a formless group. Hotter still, moving faster, we separate, the energy of our motions carrying us out into broader space, spinning apart to encounter each other in soft glancing collisions before the call now colder, cool down! slows us to regain sliding contact as we condense to twos and grouplets that contract colder now! to a jostling mass that searches for the triangular handbonds as it settles even colder now! into a quivering solid whose geometry does indeed evoke a crude image of molecules arranged in orderly array.

        So goes the dance, and it's always a struggle to keep it from spinning out of hand in deliberate collisions, even with careful forewarning. Or rather, it’s an opportunity to halt the process, talk about deportment and how responsible they must learn to be to learn to do science, and ask whether they want to try again. When they do, I mostly relax; the dance is coherent and meaningful and fun, complex enough and accessible enough to take pride in doing well as a group, and two or three cycles leave us doing the full cycle with rude grace, before I must dismiss them.

        When we meet again, I drop a pinch of copper salt in a glass again, and as it dissolves we play with blue balloons, at least in our imaginations. We are dancing molecules of liquid water, swirling in the room, sweeping up a heap of balloons until they all bounce suspended, off the walls and ceiling and our bodies, and briefly on the floor before being taken up again. We realize we can keep only so many balloons in the air, no matter how frantic our motions. As we slow down, we can't hold up so many; some fall on the floor, more do as we slow more, until they stir in clustered heaps; so also do they if we swirl out of the room one by one, leaving only balloons at the end. In this way we model in molecular terms what it means for a solid to dissolve in a liquid and be recovered by cooling or evaporation; as well as basic concepts of saturation and thermodynamic behavior.

        We will refer to these images, of solution and states and their changes, many times in the future, as we try to comprehend phenomena we observe -- harking back often to this concrete and kinesthetic introduction, as much a grounding of ideas in their own experience as I can manage. But from here it's a straight road to the immediate prize, and we start right out in what's left of our hour. I bring out the gallon jug of saturated copper sulphate solution, permanent markers, and enough small plastic petri dishes for them each to write their initials on one’s bottom. When the dishes are settled in two large plastic containers, I fill them carefully, show the kids how to refill them, and leave a full beaker of solution.

        Over the next four weeks we will grow small seed crystals, isolate them, and nurture them as they expand into sharp-faced hulks bigger than my students' fingertips before we run out of space and solution and patience. We are done with theory for now, and the practical details are straightforward: how to handle tweezers to turn the crystals as they grow, how to slow evaporation by a lid on the big container, how your crystal grows funny if you forget to keep your dish full. As our sessions go on this tending becomes routine, and our immediate interests shift to caterpillars, lizards, and autumn mushrooms. Still they must tend them each day or two, learning responsibility for their desires. By the time we are done, they deserve their prizes -- blue gems of their own husbandry, some first-rate by specimen standards, all gleaming with authentic facets and accomplishment, all the more delicious for my accompanying note to parents explaining the gems' somewhat poisonous nature and modest precautions.

        The crystals are a first fruit of their power to understand, a prize won by translating science into modest technology, for no lesser purpose than beauty and joy. Such are the meanings that throb within the act, and radiate invisibly through consciousness and life. Though we've gone about this unpretentiously, there's no harm in making the moral explicit, for it's basic enough for a seven-year-old to grasp. At the end I lead them to consider our activity, rather in the way we've considered the gross changes of water and our own behavior as molecules, by defining terms and describing their relations. To know is science, to apply is technology; we took our learning and learned to make something wonderful, of molecules in ordered array.

[A Multi-faceted Lesson]

        By the time I greet them as fourth-graders at the start of a new year, we have played with so many other sorts of wonder that they respond to my crystals' glittering display with renewed delight and more confidence. To the specimens still on the table from the preceding second-grade class, I've added a few more fragile and gaudy, and two rounded lumpy ones, dull brown, that draw little attention. After reminding them of the stern rules of handling -- one at a time, no contact between specimens, no clunking or sliding, no prying, pass them clockwise -- I leave my students to their own pleasures for a while, moving round the tables to answer questions semi-privately.

        When the sixty specimens have cycled round, or nearly, I clap for attention and recall what they learned, defining a category that embraces much of the array. The amethyst clusters, cubic flourite crusts, drapes of drusy quartz over sandstone, and more -- all these formed in the same way they grew their own crystals, deposited on surfaces from salt-laden waters, though more often by cooling than evaporation (we recall the balloons falling as we slowed.) Yet there are alternatives. I single out the massive, perfect cube of pyrite, admired by all; and then a pale, hand-sized chunk of altered sediment, sparsely spotted with dark squares and corners. Looking closer, we recognize them as smaller pyrite cubes, dark-skinned by some flaking mineral, peeking shyly forth from within the solid rock itself. And I recount the slow drama of their formation, the molecules migrating through infinitesimal interstices in the hot rock, the cubes expanding over the aeons, forcing the matrix molecules away, growing as large as the one on the table, larger, as the living rock-world beneath us continues its slow metabolisms.

        The ball-and-wire model still sits in the center of the table, evoking the mystery of structure, reminding us how each definite shape before us is formed by molecules in simply-ordered array. My students are bursting with questions and observations, about this weird specimen or that, but I tell them to stuff it all back for now, I've got something else in mind. "You're older now," I growl, "and I'm going to be tougher on you this year." They look wary. "Not mean, and no tests. But I'm going to ask you to think more difficult thoughts, work harder with your minds, because they're growing too, and they'll grow stronger if you exercise them more. So I'd like you to help me with a problem. What do we have here on the table?"

        "Rocks?" ventures Nathan, quicker to speak than to think.

        "Come on!"

        "Crystals, crystals!" an eager chorus corrects him.

        "Sure, everybody knows that, right?" They agree. "So here's my problem. I've got a friend coming to town next week, she's a giant blind slug from Mars. She knows that Earth has a lot of interesting things, out in the country, in nature, and she's heard that crystals are one of Earth's real treasures. Do you agree?" They certainly do. "So she really wants to go out in nature and see some -- no, she can't see, I mean feel some. Only she doesn't know how to recognize a crystal. Can you help her?"

        "I could take her to the rock shop," offers Gail.

        "No, I mean help her mind. How can she tell when she's feeling a crystal? What is a crystal?"

        They fidget silently, confused. Their sense of what I'm asking for is strong but vague, no one knows where to begin.

        "Look around the table." We all look. The treasury of variety before our eyes is overwhelming in its detail: so many colors, sizes, shapes, textures. "What is a crystal?" The tension deepens. Both to focus and to relieve it, I pick up a single crystal of transparent quartz, large as a fourth-grader's fist, its perfect tip bright with light; and one of the dull-brown lumps; and hold both out on my palms. "Now look. You all know which one is a crystal, right?" They agree. "I mean, is there anyone here who can't tell the difference between a crystal and petrified dinosaur poop?"

        There is general merriment, as most of them are familiar with how I work; but the three new students are uneasy. "Is that really dinosaur ... poop?" asks Sara hesitantly -- asking actually the deeper question, is it really okay to talk about such things here?

        "Absolutely!" I affirm. "It's petrified poop, fossilized feces. And I'm pretty sure it's from a dinosaur. It's turned into stone now." I drop it near Sara; it clunks convincingly. "But we could still use it to find out what she ate." My students regard the lump dubiously, expectantly; finally someone asks how. "You cut a thin slice with a rock-saw, and grind it so much thinner that light can come through it. Then you look at it with a magnifying glass to see if you can recognize leftover parts of animals or plants."

        Though the lump still looks disconcertingly like dog-shit, Sara reaches for it, against the rictus of revulsion still quivering on her lip. For my solemnity, as much as the idea, has made it clear that this is not just some weird old piece of shit, but sacred shit, pregnant with discoverable mystery. Welcome to my class, little sister my heart sings; but I snatch the lump back and growl, "Not now! We'll talk about it some other day.”

[Striving for Definition]

        “Now let's try again,” I say. “Remember the blind Martian slug. What's the difference between a crystal and a piece of petrified poop?"

        "It's transparent," Kevin says, too quickly.

        Looking beyond my hand to the display, Melinda corrects him: "Crystals have pretty colors."

        "So what? The slug can't see," says Josh.

        "You're right," I agree. "But even for us, is this how we recognize a crystal? Do all crystals have pretty colors?" We look them over critically. All colors are pretty, in a sense; but so many are here, and many so dull and ordinary, that color is clearly not what we use to identify crystals. "Try again. Think about touch. How can my friend tell?"

        "Crystals are sharp," says Josh. "And they come in clusters."

        "So my friend's going to hug a porcupine in the woods, and think it’s a crystal?"

        "But they are sharp," Josh persists.

        "So maybe you've got some useful information for my friend. But it's clearly not enough, right? And maybe you haven't said it just right. Is 'sharp' really what you mean?" We look the crystals over again. A few kinds are spire-like, but most are relatively blunt.

        "They have points!" two or three say, almost together. I ask the class whether this term seems more precise; they agree.

        "Okay. So that's one point for my friend. But it's not enough. Try harder."

        There is actually a little silence, before Aaron begins uncertainly, "Crystals are cubes...?" He breaks off, confused by his obvious error. The table is littered with distinct shapes that he has no names for, each more complex than the pyrite cube before him.

        "He's wrong," I agree with the class. "But is he just wrong, or is there something useful in what he's trying to say?"

        They puzzle it. "Crystals have shapes..." someone begins.

        "What sort of shapes?"

        "They're smooth!" says Kevin, again too quickly.

        "So my friend will think an egg is a crystal. Is 'smooth' really what you mean? Look again."

        "Flat! They're flat, they've got flats!" Their clamour sounds assured.

        "Okay, that's point two. I mean point three. Because there's something else between the points and the flats. Look at the quartz crystal. What else does its shape have, besides points and flats? It's right between these, can you see it?"    

        "Broken places?" observes Seth, accurately.

        "Well, yes, but something more important."

        "Lines?" Melinda quavers. "They all have lines?"

        "That's right. Nice curvy wriggly lines, right?"

        "No, the lines are straight."

        "I can't hear you. What kind of lines?"

        "Straight lines," she says loudly, her shyness fled.

        "Why are there straight lines? What do they do?" With the quartz crystal I goad her until she can say it explicitly: a line forms where two faces come together, and connects two points, or at least starts at one.

        "Okay. So you tell a crystal by its shape: a crystal has points, straight lines, and flat faces. Is this enough of a description, can my friend use it to recognize the crystals among the poop and everything else in nature?" They are happy to agree. There's something satisfying about it, it seems to cast a sufficient net over the variety we see; and they're relieved to have reached a goal. I tease them: "It's a really complicated description, right?"

        "No, it's simple!" Josh says happily, relaxing.

        "So if it's so simple, why did it take so much work to drag it out of you all?" I let the unexpected angle sink in for a long moment, before adding, "I mean, that did take work, didn't it? Did it feel like easy work?"

        "No way," says Melinda, "that was hard!"

        "Who else felt it was hard work?" Hands shoot up, they are glad to confess the difficulty together. "You're right, it was hard. And look what all that got you: just one simple description. Was it worth the work?"

        Marilyn says "No," somewhat haughtily; but the kids all know how she acts, and a chorus of assent ignores her.

        "That's right. It's a good description, a useful description. But let's talk about what kind of work you did to get it. Did you actually learn anything new about crystals?" They ponder the question. Whatever they sense, they can't put it in words. "I mean, you knew what a crystal was, right? Is anyone here better able now to tell the difference between a crystal and something else?" Heads shake. "So you knew something, but you didn't know what you knew. All that work was just to find out what you knew already, just to see it clearly so you could say it well to someone else. Is that right?"

        They nod uncertainly, trying to digest this. I take them back to basics. "How did you learn what a crystal was in the first place?"

        "By looking at them, of course," says Marilyn snippily.

        "By observing them," I correct her gently. "Was it enough to observe just one crystal, to be able to recognize them?" This logic is easier to follow, and they fill it out: they had to observe many crystals, many kinds of crystals, before they could be sure of recognizing crystals as such, rather than just as quartz crystals, pyrite cubes, etc. "But you still couldn't tell how to recognize them, tell the idea of their shape, until you thought about your observations, right?" I lead them through the responsory. "Were your first thoughts accurate?" No. "Did you have to try several times?" Clearly. "Did you have to keep checking back and forth, between your thoughts and your observations?" I could go on, but this is enough. "And it took you so long, it was such hard work, just because you're all so dumb. Right?" A few with good reasons to hesitate do so, but most are glad to disagree loudly.

        "Now listen carefully. What you did is one of the kinds of work that grownup scientists do. They do it just the same way, and it's just as hard for them, and they feel just as good when they get it done right." I let it sink in. "Think I'm kidding you?" Not even the smart-alecs nod.

        "Now think about your result, the description you came up with. It's simple, right? But was it easy?" The chorus comes quickly. "So simple doesn't necessarily mean easy. Thinking simple, useful thoughts is hard work. Now, I could have saved you that work. I could just have told you that crystals have points, edges, and faces. That would have made sense to you, and it would have taken about ten seconds. Right? I probably could have told you a hundred things about crystals in the time you took to work out that one. If I had, do you think you would have learned more?"

        Some don't, some do. I lead them to draw out their opinions, until it becomes clear that we're talking about two different kinds of learning. They would have learned more facts, but less about how to figure things out themselves. I offer them metaphor, talk about mental muscle, how it grows through exercise. "So which is more important, to learn more facts or to get stronger in your mind?" I have slanted the choice to get the response I want, and I affirm it as best I can. "That's right. I've told you before, that an hour with a good book will teach you more than any science program you can watch on TV. Now listen. Any hour you spend like this -- trying to figure something out for yourself, and coming up with one simple idea you can explain to someone else -- is worth more to you than a book and a TV program put together. Even if you think you've learned hardly anything. Please believe me!"

        If I leave them no space to think about how important facts are, or about the deluge of facts they learn from me, it's because I don't care about balance here. This little is all I can do to prepare them for what they will face so soon -- the inexorable mill of tests and grading, that defines good learning by numbers and will measure only how well they remember facts and apply simple concepts, and not the deeper work of mind, nor nourish this.

        Ah well. I am mostly beyond such railings, I have tried to put the thoughts and feelings to compost in my work. But sometimes they erupt, when I realize anew how little of the deep work I recognize, how easy it is to avoid, how scanty my efforts are to help with even what I might. Yet how can I tell my students what can only be learned by reflection on one's own experience, like the nature of crystals? Or provision them for what they face?

[Crystals Are Unique]

        Fortunately, they have their own resources. While I've been pontificating, Josh and David have been conferring at their corner of the table, in irritatingly-audible tones. When I pause, they spring it on me, their arms waving so energetically that I must call on them. "What about a box!" Josh exclaims triumphantly.

        "What do you mean?"

        "We think your definition's wrong," says David. Amused that this makes it my definition, I beckon him to go on. "Like a box, the slug could feel a box and think it's a crystal. Because it has points and edges and faces."

        "No it doesn't," says Melinda, rallying to defend me as much as the idea. "Its edges aren't really sharp. The corners aren't really points either. A cardboard box," she adds -- perhaps as the idea of a box that would have sharp edges strikes her.

        "Well, a Lego block then," says David.

        "You're both thinking good thoughts," I say diplomatically, before telling David, "And you're right, there's a real problem with the description we came up with. But it's not a problem for my friend the slug, because she just wants to use the description outside. Look around the room. Do you see any straight lines?" Of course they do. "Just a few?" Too many to count. "Now look at the surfaces in the room." They get into it, craning their heads in all directions. "See a few flat faces?" The familiar contours of cabinets, tables, walls, fishtanks, doorjambs unfold themselves to our eyes in sudden identity. "And the straight lines form..."

        "Where the flat faces come together!" cries Betsy in delight, as the crystal geometry embraces us all. I am hardly aware of it consciously at the time; but perhaps some deep resonance of wholeness throbs through us as we sit around the crystals in the crystalline confines of the room, crystallizing our understandings. What is the learning that might dwell here?

        But I rush past the moment, determined to sum up. The institutional architecture makes it easy. "Almost every surface in this room is a flat face, every line's a straight line, we're surrounded by them, the slug would go nuts. Think of your own room at home, isn't it mostly like this? That's how our people make things -- though you should know that in some other cultures, other tribes, they like curved lines and unflat surfaces more than we do.

        "Now go outside in your mind, go into the hills." I stick up my arm, fingers spread in a little crown. "Look at a tree. We say it comes straight out of the ground, but really, is its trunk a straight line?" And we are into responsory again, modeling a process of inquiry to follow to conclusion, as they agree it's not straight and I lower my arm. "Look at its leaf." I wiggle my outstretched hand. "Is it really flat?"

        "Leaves are flat," says Julien, staring at my knobby knuckles.

        "Look with your mind." I let my fingers droop suggestively. "The leaf isn't flat, it's ..."

        I let the words hang, until Meadow says, "Thin. It's thin, not flat. But some leaves are nearly flat."

        "But they aren't really flat. I can make a leaf flat" -- my other hand mimes an iron pressing the leaf -- "but that's just a person making things flat. Now look at the ground, at the moss, the rocks, the plants. See any straight lines or flat surfaces?"

        No way! But Josh is casting about. "How about a pond," he says, "that's a flat surface."

        "Look at it more closely. Isn't there a bit of a breeze riffling the water?" I'm lucky he doesn't offer me a spider-web, though I might still use the microscope to show him its irregularities. For I'm cheating a bit, in a way that I trust does no harm, to pounce on the neat conclusion. "So think about it. All around us we've made straights and flats, but if you go out in nature you can't find any. Except crystals. Crystals are the only thing in nature with straight edges and flat faces. They are unique. That means there's nothing else in the world like them.

        "Now think of the first people who ever noticed crystals, a hundred thousand years ago, maybe a million. You're climbing through grass on the hill, hunting for food, turning over rocks, and suddenly you find ... this!" Light glints from the big quartz point as I flourish it around the table. "Isn't that the most amazing thing you ever saw? I've never seen anything like it! It's so special it must be from the gods, it must have a very powerful spirit in it. I'm going to wear it around my neck, between the bear claws." I stride around the table, dangling the crystal from my throat. "So what have I got?"

        "A necklace?" ventures Sara.

        "Be more general," I say, wiggling the crystal until it dawns.


        "Uh-huh. Isn't this a jewel? That's where people got the idea of jewels, from crystals. First we just used crystals, then we learned how to take pretty stones and grind them into the shapes of crystals, with flat faces and straight edges and sharp points. Then we learned how to grind them into even more complicated shapes, and make them even more perfect than natural crystals. But I still like the natural ones best!

        "Now listen. I have to warn you about something. As you grow up, you're going to meet people who have some strange ideas about crystals." What an understatement, in Berkeley 1989! "Some people think there are spirits in crystals, that speak to them. Some think that if you sleep with a crystal like this" -- I pick up a lucent garnet -- "right next to your heart, you will feel more love. Or if you have stomach trouble, that holding this other one can make you better."

        "Can it really?" asks Sean.

        "I'm pretty sure it can't. Some people do get better that way. But people can help themselves get better in so many different ways, that I think the magic, the power, is in the person, not the crystal. I'm not absolutely sure, but I wouldn't believe the other unless someone could prove it to me. I'm not interested enough to try to check it out for myself -- though I might be, if I had stomach trouble. But a real scientist never closes his mind completely to any idea, unless he's sure he has proof that it's wrong. And even then he should be willing to think about whether he's mistaken.

        "So I'm telling you something complicated, about how to listen when people say strange things about crystals. Most of the strange ideas you'll hear are probably just nonsense, but some might have something true in them. Remember the first things you said about crystals, that weren't really true but pointed in useful directions? There may be more to learn about crystals than we know yet. But you've got to be very careful not to just believe the ideas you hear. You've really got to think about them, and think about what kind of proof people have for them."

        "That's silly, that spirits live in crystals!" says Heather.

        "I think so too. I think it's just a superstition. But I don't treat people who believe it as if they're jerks.”

[Dealing with Feelings]

        “Now listen. I've told you this partly to warn you, and partly so you won't confuse me with someone with weird ideas when I ask you to pick up a crystal. So pick up a crystal." They boggle at me, taken aback by the shift. "Pick up a crystal, right now! Pick up one you like, hold it in your hands." They scramble for the prizes. "Quit grabbing! There's enough for all."

        When they're settled with their gems, I lead them on. "Now look at your crystal. Forget about everyone else, about me, just look at it. Look at it and feel your own feelings. What do you feel?"

        "It's rough," says Sean in wonder, rubbing the massive fluorite cluster he's chosen, for its faces are pitted and rugged with hundreds of small cubic shapes echoing its larger forms.

        "I don't mean feelings in your body, forget that. In your mind, what are your feelings, your ... emotions," I end lamely, embarrassed at the abstraction. "Look at your crystal. Let me guess. You feel angry, right?" They're sure they don't. "Then how do you feel?"

        "It's pretty," says Catherine.

        "No, mine is beautiful," corrects Melinda, her tone so rich with reverence I can hardly bear it.

        A long pause. Finally Josh says confidently, "It makes me feel happy;" and then the feelings come tumbling out as I call on the sudden forest of hands, as fast as I can take notes on the board. How expressive they can be, and how precise, when they testify! "I feel like before a party," says Heather, and another: "I feel wonderful." Marilyn will say no more than "it feels nice" even with coaxing, with so little enthusiasm that she's clearly just mouthing the party line. But Aaron surprises us with his quiet assertion, "It makes me feel sad," unmistakably sincere.

        I long to ask him why, and also to correct their dangerous usage of "it makes me," but dare not break the spell. And so earn my reward, for now silent Stephanie says shyly, "I get goose-bumps" -- evoking something too deep to name, and telling me what a fool I was to instruct them to ignore their bodies in reaching for their feelings. How many of their hearts beat slightly faster, and how many slower; and is it not science to ask why as well as how?

        A dozen testaments have made the field of feeling complex by the time Kevin declares, "It makes me feel proud," fondling the perfect pyrite cube. But I have had enough.

        "Enough! Now think about it. Did your crystals really make you feel your feelings, or is this just how we talk? The only thing you know for sure is that these are your feelings, they're the feelings you have when you look at your crystal. But who do you think makes you have your own feelings -- the crystal or you?"

        Themselves, they say, with enough confidence to placate me for now. I have slanted their election, skirting deep issues here. But there's so much nonsense abroad now; and so many deep reasons to lead them to recognize their own sense of responsibility, to sort it out and emphasize it and save it from all the contrary teachings and forces that erode it in their lives, in this age.

        "All right. Now let's look at the feelings you had. Was it very easy to figure out what you felt?" A few think so, but most do not. "Was it as hard as figuring out what you knew about crystals?" They're sure it wasn't; but they decide that the difficulty was similar in some ways. I wonder whether it was less because no one questioned their answers, and because they did not need to agree together. But I say only, "You're right. Sometimes it can be as hard to know what you feel, clearly enough to say it to someone else, as to know what you know. If you practice, you'll get better at both."

        I turn to the data on the board. "So, who felt excited?" Many hands wave. "Who felt lonely?" Only two. "Who felt caring?" As I lead them down the list, I order its randomized elements into a progression in my mind, or rather a cycle that leads back through itself, and marvel at how much it encompasses. My students felt cold, sad, lonely, scared, weird, dizzy, climbing a mountain, rollerskating, at a party with lots of lights, excited, proud, rich, safe, caring, interested, curious, happy, longing, with goose-bumps. What rich diversity! Does it simply measure the way we use the crystals as mirrors, to reflect the many faces of our own being? Even so, how admirably they serve the purpose!

        Is there any other thing in the world that could evoke so rich a range of response from us? Surely crystals are unique. And yet any other thing -- a puppy, a geyser, an ant, a dull rock -- can evoke so rich a response and more, if we bring our selves truly to apprehend it. And so I lead them to study the wayside weed, the discarded appliance, the scratch on their hand, with no less reverence and care than we invest in crystals today. Nothing is negligible, everything connects to all; the lessons of science unfold without end from the least place we choose really to pay attention, as I have learned -- as do the lessons of the self, as the old teachings tell.

        So what's special about crystals? I really should inquire into this with my class, if only to introduce them to the idea of well-controlled experiment. We'd put fine specimens of ten different crystal species into identical, closed, numbered containers; find places where we can be separate from each other and undistracted; spend three minutes holding a container, feeling what we feel and finding words to make a few notes; exchange containers and repeat until we each have been through all; sit down together and compare our notes on each container; try to decide whether different patterns of feeling appear attached to different containers, or whether our attachments of feeling seem random; and only then open the containers, to identify which crystals evoked which patterns of feeling, if we do find any. This might not settle the question of whether crystals themselves exert any sort of independent influence upon us, but it would be  a good start -- and even more, a good introduction to critical thinking itself, as well as to particular elements of scientific methodology.

        Meanwhile, we've got more vital learning to follow through. "Now think about what you've all said together," I say, after we've run through the list of their feelings. "I trust you've been listening to each other. Is there anyone here, who didn't have more than one feeling?" No one volunteers. "Who had only two?" Only a few stop here; most can identify at least four feelings in themselves, or think they can, which may come to the same thing. "So everyone's feelings were complicated. A crystal is simple, but your feelings are complex. Let's hear it. Your feelings..."

        "...are complex," they chorus, in obedient custom.

        "So did everyone have the same feelings?" Certainly not, they clamour. "Did any two people have exactly the same complex of feelings?" After confirming what complex means as a noun, we check it out, among the least-complicated: not even here. "So no one else had exactly the same feelings that you did. Your complex of feelings is unique. Let's hear it, get used to the words. 'My complex...'"

        "My complex of feelings is unique," they repeat together.

        After summing this line of our research, I link it with another. "So who had a feeling that was only their own? Who had a feeling that no one else had?" No one, they decide -- not even the goose-bumps girl, for several were sure they'd felt her feeling even though they hadn't noticed the bumps. Did they really; or were they 'simply suggestible', i.e. empathetic? It comes to the same thing. "So whatever you feel about crystals, there are other people who feel the same way -- maybe not many, but some. We can put this in a simple way: My feelings are shared."

        I beckon; they repeat. With an older class and more time, we could go on to sharpen this conclusion. We'd each make a list of our feelings, after hearing the whole list; and then follow a patient combinatorial process of comparing our lists, to find out how many pairs among us had not a single feeling in common. This would be a crude empirical test of the dual proposition: No one is a complete stranger to me, nor am I to anyone; I share some feelings with each other.

        But for now, three morals are quite enough. I chalk our conclusions on the board, beside the raw data I have helped my students to interpret: My feelings are complex. My complex of feelings is unique. My feelings are shared. "Now look where we've gotten to. First we tried to describe these crystals. What we saw together was so complex, but we looked carefully and came up with three simple ideas to describe it. Next we tried to describe our feelings about the crystals. What we felt together was so complex, but we looked carefully and came up with three simple ideas to describe it."

[Playing with the Loose Ends]

        Enough, I must settle them to leave. "So look at your crystal specimen, one last time. Let's be sure that the description we worked out is really useful here. Check the crystal you picked, to make sure it's really a crystal. Does it have points, edges, and flat faces?" Most of the children comply and nod routinely, with no inkling of the point -- but it catches two sharply, as they struggle to describe what they realize.

        I move to Sara, to help her. The slab of violet mica she holds is luscious, its glistening surface studded with small tourmaline sections. I go over it with her aloud. "Flat faces, hey, big ones. Edges? No, these are ragged -- except here, see where this one split straight? And points? Nope. No points. My, isn't that interesting!" I move to Betsy, who picked a silky chunk of asbestos, its densely-packed parallel threads frayed at top and bottom. I fondle its irregular contours, poke myself with it. "No faces. No points. But edges? Lots of edges! It's just a bundle of edges." I tease a long thin fiber free to show them.

        "So are these really crystals?" I ask, and lead them through the brief logic. Clearly, they fail the terms we defined. Yet one specimen fits one-third of our description, and the other fits two-thirds. Moreover, something about them reminds us insistently of crystals, they are both so ... structural. What can we make of this mystery? Is our definition crippled?

        I pick up the perfect pyrite cube, that textbook model, and introduce dimensions formally, tracing them with my finger: length, width, and thickness or height, that makes three. "Now check out your specimens. Who has three-dimensional crystals?" Everyone decides they do, except Sara and Betsy. Aha! We look at their specimens. Sure enough, something's weird. The mica chunk is longer and wider than Sara's hand, but hardly thick.

        "But it's some thick," she observes.

        "Well, yes," I concede. "But you haven't quite realized what you're holding. That's not a single crystal, it's a whole cluster of crystals. Except they don't look like a cluster, because they're flat." Reluctant to whittle my prize down too fast, I take care with the razor-blade, and succeed in separating a mica sheet so thin that its color vanishes and I can hardly hold it. "This is probably a hundred crystals I'm still holding here, maybe a thousand. They still do have some thickness, but they're really very thin. You can see why we call it a two-dimensional crystal. Because the cluster lies together like pages, we call this a 'book' of mica."

        I take up Betsy's asbestos chunk. "This looks like it has three dimensions. But if we look closer" -- I strip another fiber -- "we see length, but hardly any width or thickness. Now if I tell you this fiber is still just a cluster, that the asbestos crystals are very skinny and there's a thousand in the fiber, you see why we call them one-dimensional."

        So two-dimensional mica fits two points of our description, and one- dimensional mica fits just one! This strikes Josh as a pleasant consonance, and though lunch is pending I linger briefly to extend it. There's no time to go into the molecular details, of how mica sheets and asbestos threads hold together, and why they do so only in some dimensions. But they must clearly be included in our definition, and we ponder how to revise it.

        Could there be other crystal forms that satisfy only one or two of its three terms, but a different one or two? They decide that one that had only points, but not edges or sides, would be so tiny it was invisible; but there's no time to discuss the idea of a point being zero-dimensional. And a crystal with only a face, and neither points nor edges? They argue; some think this would be a better description for mica, and others say no, mica does have edges even if they're rough, and a crystal without edges can't really exist. They are right, thinking of flat faces; but someday I'll blow them bubbles, all surface and almost no thickness, and we'll talk about whether the globular smithsonite specimen that nobody chose was a crystal cluster too in its own way.

        Passing to pairs of terms, they can't imagine anything with flat faces and points but no edges, and neither can I. "But wouldn't a broken asbestos crystal have a point too, not just length?" asks Melinda.

        "We can see it that way too," I agree. "So it seems that your description was good enough after all, if we say that anything natural that fits at least two of its terms is a crystal. Congratulations!"

        Our quest brought round at last, I dismiss them to other nourishment. But Stephanie lingers, to bless me with goose-bumps. "Is there really a fourth dimension?" she asks.

        "Oh, there's more than one," I tease her. "Some scientists think of time as a fourth dimension. But the one I like is simpler, it just grows out from ours in the same way. Like this." I take the pyrite cube, and trace the edges leading from one point. "One line, one dimension. Two lines perpendicular to each other, two dimensions. Three lines perpendicular to each other, three dimensions. Then you just go on." So goes the classic demonstration: twenty seconds to share, a lifetime to ponder.

        Stephanie tries to grasp it on the wing. Much luck! Still, she persists. "Are there crystals in the fourth dimension?"

        "You mean, are there four-dimensional crystals? Oh, there are lots, and five-dimensional and six, and crystals with too many dimensions to count." I grow giddy with hunger as sun strikes through the window, remembering another lunchtime half my life ago, on the terrace at the university, sipping cold tea as I worked on a theorem in algebraic topology, my mind skiing sensuous homomorphisms down infinite nested sequences of function spaces, watching a girl with a marvelous tan. "Lots and lots of crystals. I used to play with them. They're very beautiful. Only you can't see them with your eyes. You can only see them with your mind, and maybe something more."

        "How do you learn to see them?"

        "You start by thinking about the cube and what I showed you. There's a drawing of a four-dimensional crystal named tesseract, that helps some people learn to see it. But your mind has to grow ready to see it before you can learn. When you do, you'll be able to find books and people to help you learn. So study your geometry, eat a piece of fruit every day, go play in the sun. I'll see you next week."



Some further details of the introduction

        Our experimenting begins with familiar table-salt, dissolved in a glass of water. What happened? It disappeared, it dissolved. Is it still there? We taste; clearly, it is. So what does dissolved mean? Though some use the word confidently, no one has any real sense of its meaning. Somehow the salt has turned invisible and been distributed throughout the water.

        When we do the same with copper sulphate powder, I introduce this salt by name and explain that there are hundreds of different kinds of salts. I classify them as good, neutral, and poisonous; identify copper sulphate as poisonous; and respond to their alarm at having handled its crystals by clarifying the mild degree of its poisonousness and the simple precautions appropriate. I tell them that we'll always call table-salt by that name, or 'sodium chloride', to avoid confusing our salts.

        With copper sulphate, the salt is visibly there in the water, and visibly uniform in its distribution. After concluding that it's in bits too small to see, but before discussing molecules, we return to the table salt. First we examine it with our eyes; they recognize small cubes. I focus slides of salt under the microscopes; the magnified cubes are exciting in their precision, I identify them as small crystals. I show my students how to use the focus, and let them begin to get the feel of focusing a microscope, which we'll use in so many ways thereafter. While they practice, I dry some slides of salt-water carefully. A white powder appears; and tasting confirms the fact that we've recovered the salt from the water. Looking at these slides under the microsope, we find tiny crystals again. Later, a simple molecular model of a cubic crystal, compared to the rhomboidal model, concretizes the idea that differently-arranged regular arrays of molecules give rise to different crystal shapes.

        While their blue crystals are growing at school, I encourage students to explore growing table-salt and other crystals at home; and send home handouts that tell where to buy other simple salts and how to go about growing them. Some students try growing sugar crystals, which offers special problems and delights.

        After learning our molecular dance, we put crystals of both salts into still glasses of water, leave them covered for a week, and discover that the salt molecules have become distributed throughout the water, with no help from our stirring. This helps make plausible the idea that the molecules of a liquid are in continual motion on their own. Our dance also prepares us to understand why, as a simple demonstration at the table shows later, a solid has both a particular size (volume) and a definite shape, a liquid has a size but no definite shape, and a gas has neither size nor shape of its own.

Connections with further lessons

        In introducing molecules, I pass lightly and inaccurately over the issue of energy, as I don't explore the complexities of this concept with second-graders. “Heat is molecular motion,” I have them chant together, so often that it becomes a ritual phrase, slowly illuminated as I evoke and apply it concretely on dozens of occasions during the next four years. Why does sugar dissolve faster in hot water? Why are smells stronger in a warmer room? Why do microbes grow so slowly on refrigerated food, and tadpoles metamorphosize sooner in the warmer tank? Why do butterflies wait for sun before they can fly in the morning? How does a burn kill your cells? Why do things expand as they warm up? How do heat conduction and insulation work? Each time, we think from scratch about molecules moving slower or faster, as the underlying unity to such diverse phenomena, of which we find no end.

        Crystals too figure in many subsequent lessons. We observe water crystals in the undefrosted refrigerator and snow; zinc crystals on the surface of ordinary galvanized iron; and nickel-iron crystals on the etched face of a meteorite. Mixed crystal grains are our key to identifying granitic rocks; our understanding of how they form from melt enables us to tell the plutonic from the volcanic among igneous rocks, and to estimate how slowly a plutonic specimen cooled. Crytalline veins and crusts trace the actions of water upon the land. Grasses line their leaves with microscopic crystal daggers to wound the hungry mouth; crustaceans use calcite crystals bouncing on sensitive cells to tell which way is up; birds make crystals of iron oxide to guide their flight through earth's magnetic fields. Crystal domains are vital to the phenomena of magnetism and demagnetism, and thus to the technology of magnetic media. A crystal of optical calcite introduces us to polarized light; skinny crystals do the polarizing in our sunglasses. A galena crystal lies at the heart of the original ham radio receiver; a quartz crystal introduces us to piezoelectricity, the sparker that lights the kitchen stove. An excited synthetic ruby pulses at the heart of the laser; and the computer's intricate brain grows from a substrate of crystalline silicon.

        Indeed, a crystal thread runs through every domain of science that deals with material stuff; one does well to pick it out early to follow.


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