A dramatic classroom demonstration enables a high-school girl to grasp geometry as a tool she can actually use.

EVERY TWO YEARS OF MY CHILDHOOD, I was the new kid in school. My father served in the Air Force, and each time he was reassigned the family followed. After a move, I was again the outsider facing a pack of suspicious classmates. Invariably, and to my dismay, suspicion turned to dislike after the first exam when my test scores edged those of the alpha student.

My strategy to stave off the resultant unpopularity was simple: look as nonthreatening as possible. I couldn't change my test scores, but told myself I could sit quietly, look friendly and give them little else to resent. Class was a place I sat and listened. Perhaps if I didn't stand out, my new classmates would ease up.

I kept this ultra-passive approach to learning until one day, in class, an unprecedented event shook me out of it. As ninth graders, my classmates and I had been having trouble getting our minds around geometry. Abstract collection of lines stretching on forever, containing infinite numbers of points, matched nothing we saw in the world around us. What were loci? Conic sections? Cartesian coordinates? We hadn't read these words in a math book before.

Even teaching us looked like hard work. Mr. Cook always rolled up his sleeves before class. The armpits of his blue button-up shirts were usually stained with sweat, no matter what the temperature was indoors. Early in the year he asked us, "What's the intersection of a line and a plane?" Silence. He fished a pen from his shirt pocket, uncapped it, stood on his chair and jammed the pen point -- first into the acoustic-tiled ceiling. "What's that?"

The class giggled, and the scales, as they say, fell from my eyes. The pen stuck in the ceiling forced me to see the answer. The intersection had to be a point -- or, I decided, rotating the pen in my mind, the line itself if the pen lay flat against the ceiling. I wouldn't have been able to rotate the abstract concept of a line going on forever in both directions, but it was easy to do with the pen.

I couldn't imagine getting up and sticking a pen into the ceiling myself. Yet this teacher's simple act freed me, paradoxically, from needing books -- and teachers, for the most part -- to understand math.

Mr. Cook was the first teacher to give an inspired demonstration with everyday objects in class. I'd taken part in a handful of carefully planned class activities designed to teach us lessons before. My fourth grade teachers somehow hauled a pillow-sized genuine cow's lung, sans cow, into class. I was allowed to press the spongy pink tissue with my fingers, and compare it to a color photo of a charcoaled smoker's lung. The contrast made me a non-smoker for life, on the spot.

While I didn't have cow lungs or other organs sitting around at home, I almost always had a pen, and I lived under ceilings. It had never occurred to me to use normal objects around me to discover how things worked. For me, math always had been strictly an abstract science. Equations stayed on the page in the book. The swift translation to real objects made math something I could touch and, more shockingly, something I could actually do and create. I didn't need to sit at my desk with a geometry text if I could answer my questions with a sheet of paper for my plane and the nearest pencil for my line. Infinity became tangible.

More than anything, thinking about math in this way opened the boundaries of acceptable inquiry in school for me. I could close the book and play with the concepts. Learning wasn't always going to be sitting in a chair and hoping not to irk classmates anxious to be anywhere but school. I could teach myself by doing, not memorizing.

When I reached college, I found out that great scientists learned in that same way. Physicist Richard Feynman won a Nobel prize for theory, but he always kept real-world examples in mind. Many people remember his O-ring demonstration after the explosion of the space shuttle Challenger, when he dunked a rubber ring into ice water to show the fault in the fuel system. The 32-degree water matched the temperature at launch time, and he showed that the rubber wasn't resilient and must have leaked. Feynman couldn't fathom learning without doing. During a teaching stint in Brazil, Feynman found his students primed with physics definitions, ready to spout textbook answers to "what is torque?" with no idea what these words implied about the objects around them. Recounting the visit in his autobiography, Surely You're Joking, Mr. Feynman, he mentioned one lecture he heard which amounted to a pure listing of terms, defining one new word abstractly by others, and he wondered how useful it would be for the students.

I didn't see how they were going to learn anything from that. Here he was talking about moments of inertia, but there was no discussion about how hard it is to push a door open when you put heavy weights on the outside, compared to when you put them near the hinge -- nothing! Feynman's students could parrot the definition of refraction perfectly, but they couldn't apply that knowledge to see the implication that sunlight reflected from the nearby bay must be polarized, he said.

By following Feynman's example and making math concrete, I could demystify math and science. Explaining how the world works became something personal, something that could be done by anyone. I could experiment. Sure, I'd look strange, but right or wrong, I'd learn something. I'd learned from a book about the scientific method -- observe, form hypothesis, test it, draw conclusions -- but seeing hands-on testing gave those words a life they hadn't possessed before.

I applied this insight in school a few years after Mr. Cook's class. During an idle moment in high school chemistry lab, I wondered if I could remove the hard, whitish residue from an otherwise clean Pyrex beaker. Soap didn't budge it. Persistent scratching with a fingernail worked, but messily. I didn't know where the film had come from, but we'd been studying acids and bases and how liquids at one end of the pH scale would neutralize the other. Soap, a base, didn't work, so washing the beaker hadn't helped. I found some weak hydrochloric acid and swished it around the inside of the beaker. The acid lifted the stains instantly. I was pleased with myself for daring, for taking the chance to apply it and being right.

I didn't regret adopting a hands-on approach to learning. It didn't change how my classmates saw me: I was still new, strange, different. At the end of each year, my yearbooks were still filled with phrases like "You seem like a good kid, Wish I knew you" and "Your [sic] a great kid even though you intimidate me." Nothing I did would have changed that.

I was more comfortable in the world of science. Learning by experiment appealed to me viscerally, and science depends on experiment more than any other discipline. Scientific "facts" remain merely ideas that haven't yet been disproven -- the collective best guess of humanity.

Hands-on inquiry -- the sort that I saw in a flash in Mr. Cook's class was something I could do -- sometimes carries a risk of what the military euphemistically calls collateral damage. Once, excitedly explaining how to solve a word problem about angles on a clock, Mr. Cook yanked the classroom clock off the wall and moved the hands to show us. "At three o'clock, it's a 90-degree angle," he said. "At five, it's 150 degrees between the big hand and the little hand." Mr. Cook's helpful demonstration broke the clock. It never again kept time.

Still, Mr. Cook made me unafraid to take chances and, metaphorically, break a few clocks. I might lose track of time, but I'd know a lot more about the world I was late in.