When I walked into Masha Albrecht’s geometry class at Berkeley High School last week, her students were holding hands. It wasn’t budding romance. It was math. Before I explain, I have to tell you about Masha.
I met her when I was a senior at Cornell in the 1970s. I decided to do an independent study that involved working with students in an elementary school classroom and she was the most eager, bright-eyed fourth grader in the class. Masha’s brother Bobby was in an adjacent room and the two classes were team-taught. I hit it off with both siblings, and before the end of the semester I had been to their house several times, met their parents, and spent some enjoyable after-school hours together — doing math. The three of us had delved into Harold Jacobs’s masterful Mathematics: A Human Endeavor, then in its first edition, and we tackled the mind-stretching, often funny, problem sets with gusto. “This is how math should be taught in school,” exclaimed Mrs. Albrecht.
I re-met Masha Albrecht at a conference for math educators in northern California last December. We were both speakers. Good thing her name is an unusual one because I recognized it immediately on the program, and I sought her out. And just to close the triangle of delightful coincidence: Harold Jacobs was also a speaker at the conference.
So now Masha teaches math in school the way it should be taught. Her mother would be proud. In groups of three, four, five and six, her students were creating polygons on the day I visited. Their bodies were the vertices and their arms were the sides. In this kinesthetic approach, they were learning the vocabulary and some mathematical principles of polygons. “Now shake hands with a non-consecutive vertex,” their teacher requested. Later she told me, “They don’t get ‘non-consecutive vertex’ unless we do it this way. Now they’ll get it and remember it.”
What a teacher! Actually, I had gone to the class to see an entirely different class project she had told me about. She had decided to use hubcaps to illustrate reflexive and rotational symmetry. Here is a quick a primer on symmetry so you don’t have to wait for Loreen’s new book (see her post of Sept. 21st). It is my hope is that once you learn about this, walking past parked cars will never be the same.
If one half of an object mirrors its other half across an imaginary line, we say it has reflexive symmetry. Think of a human face, at least an idealized one. The imaginary line is the “line of symmetry.” Whether or not an object has reflexive symmetry, it might (or might not) have rotational symmetry. Imagine rotating the object partway through a full 360-degree rotation. If you can get it to a place where it looks identical to how it looked before, in the same exact orientation, it has rotational symmetry. If you can rotate it into three positions that look identical, it has three-fold rotational symmetry. Four positions gives it four-fold rotational symmetry and so on. Clearly, human faces do not have rotational symmetry.
But start looking at hubcaps. Do you see reflexive symmetry? Sometimes, always or never? Do you see rotational symmetry? Sometimes, always or never? Do you ever see both in the same hubcap? Are there hubcaps with neither? (Disregard scratches or imperfections.) The answers may surprise you. They sure surprised me. In fact, I now think hubcaps are way cool, way beautiful and full of subtle surprises. (I don’t know why it took so long for me to realize this, but — as with so many epiphanies — a teacher was the catalyst.) And here’s just one of the many discoveries that I, and Masha’s students, made: there are hubcaps with what I’ll call “dual foldedness.” This could mean a pattern that is neatly subdivided (imagine five rays emanating from a central disk, with each ray subdivided into three branches). That’s interesting enough in an orderly sort of way. But imagine a cap whose central disk has fivefold rotational symmetry and whose periphery has six-fold. Now that’s cool! Think about it, draw it, or go out and look for a hubcap bearing it (try a Toyota Prius -- some years) and decide whether that cap as a whole has any symmetry at all.
Devoted to finding real world applications to help Berkeley High students understand mathematics, Masha had a great idea. Two, actually. One was to connect “dual-foldedness” (like the 5 vs. 6 example) to polyrhythms in music. She didn’t actually end up exploring this with her students, but she did ask them to start taking photos of hubcaps and putting them into a grid she'd created at the back of the room. They would identify the photo’s symmetrical properties by pasting it into the appropriate cell of the grid. Across the top, columns were labeled for foldedness — 2-fold, 3-fold, 4-fold, etc., up to 8-fold. The rows were labeled “Reflexive,” “Reflexive and Rotational” and “Rotational But Not Reflexive.”
After a week, the grid boasted merely one hubcap picture and Masha was ready to take it down. “The project didn’t light a fire under them,” she told me on the phone. Later, in an email, she wrote, “I think once they are walking down the street all thoughts of academics are far away. I myself continue to be amazed by the variety of symmetries in hubcaps.”
She did invite me to see some lovely posters the same students had made to illustrate geometry in the real world — seashore organisms, springtime flowers, quilts and so forth. And indeed they were lovely posters, caringly created. But to me, the hubcap project was a little different because it required discovery in the real three-dimensional world using the students’ own eyes and minds, rather than research from second-hand resources like books or the internet.
Masha didn’t seem surprised at the disappointing level of participation of the hubcap project. “I will defend my students,” she wrote in another email. “I think the school environment doesn't connect to their world, so by high school they have stopped bothering to believe in any connections.”
Wow. What an indictment! It’s not hard to see how the drill-and-kill/teach-to-the test/scripted-lesson regimen that so many schools now call education would drive out any hope of connecting math to the real world. Did Masha hit the hubcap on the head with that conjecture?
Let me propose an experiment. Teachers of intermediate grade students may be able to test Masha’s statement if they can break away from the stranglehold of teaching-to-the test long enough to try this:
I believe the symmetrical properties of hubcaps are accessible to 4th or 5th graders, who have had 4-6 years less schooling than Masha’s high schoolers. So how about teaching them about symmetry and trying a similar project? (You can find a short exposition on the subject in my book G Is for Googol on the page “S is for Symmetry.” Masha likes the superbly untextbooky textbook Discovering Geometry by Michael Serra.) Your students could draw the hubcaps in the grid if they don’t have cameras. Would a project like this light a brighter fire under 4th graders than it did under 9th and 10th graders? I’d love to know. If your students can shed any light, please let it shine in the comment section here.In the meantime, I’m going out to photograph more hubcaps.