I always had trouble with math. From the second grade on I can vaguely remember falling behind already. My grouchy second grade teacher, one year away from retirement, didn’t help me feel anything but stupid or lazy. By the time I reached the fourth grade in 1963, there was talk of the “new math.” Parents were frustrated because they didn’t know how to do the new math, so they could not help their kids with homework. In the fourth grade, no matter how hard I tried, I could not do any better than a D grade. This with the fact that I had a late birthday (October) that made me younger than most of my peers, and the fact that I was the smallest kid in the class, wasn’t great for my self-esteem.

So I asked Miss Zamanegian if I could repeat the fourth grade. She said no. She pointed out that I was great with reading and writing, that I always did excellent art projects and that I was the most musically gifted student in her class. She was kind, she had me brush back my tears, she told me to stick with it, work hard, and I would get better with time.

I just barely squeaked through fourth grade as far as math was concerned. I had by then grown to hate it, felt that it gave me a headache, and that I’d never understand it. By seventh grade I was slipping very quickly, D grades all year, many lower marks on quizzes. I had no idea of what this new word “integers” was supposed to mean. Today I know I would have done much better had the teacher reminded us every once and a while that we were learning a language and that it was important to know the procedures to correctly solve a problem. Instead he would just talk at us from the board and write problems, showing us how to solve them. It felt like half the class didn’t want to say they didn’t understand when the teacher asked us; we were ashamed to be embarrassed and let any one else see how stupid and inept we were. But I shall return to my struggles—and eventual overcoming of obstacles with math and high abstract thinking later.

From around the fifth grade I decided that I loved astronomy. I loved looking at the stars, got star charts and books, and began being able to recognize the planets in the night sky. I would look at pictures of galaxies and nebulae and dream about someday being able to use powerful telescopes to see them. I decided that I wanted to be an astronomer when I grew up. My powerful idea was a deep understanding of gravity and planetary orbits. I could “see” it (through illustrations in books) so I could understand it. The picture in my mind of the moon orbiting the earth, the earth orbiting the sun and the other planets following different orbits was the easiest thing for me to understand. I didn’t realize until thinking about his now over forty years later, that I had a very powerful understanding of gravity, deeper in fact than my sixth grade teacher.

The sixth grade teacher invited me to give a couple of talks about gravity to my class. This came about when he was talking one day about how the moon might have been created. He talked about how it may have been a smaller object in space that was captured by the earth’s gravity. Or the theory that it might have formed at the same time as the earth, and because it was smaller, got captured into an earth orbit. When I raised my hand and explained another theory, that the moon may have actually been yanked out of the earth, leaving the basin which is the Pacific Ocean, the teacher was amazed and asked me to come up and explain it on the blackboard.

The first picture looked like this:

It showed the earth, millions of years ago, when it was believed to be a mass of molten rock and metal.

The second illustration:

Supposing a comet or other large body came close the molten earth and through the force of gravity pulled a large molten chunk of material out of the earth.

And then was pulled out into space until the force of earth’s gravity captured it and pulled it into orbit. Eventually, both the moon and earth cooled, but with one major difference. The moon wasn’t large enough to have gravity as powerful as the earth—it was in fact only one sixth of the earth’s gravity. While the earth and moon cooled and hardened over millions of years, the earth’s gravity was strong enough to capture an atmosphere, while the moon’s wasn’t. Because it either never had an atmosphere, or lost it’s atmosphere millions of years ago, many more meteors smash into the moon than the earth. Most meteors that enter the earth’s atmosphere are burned up by the friction caused by air—the shooting stars that we see at night.

Later in that unit I was again asked to explain how the planets revolved around the sun. It was again a very simple idea to me to imagine each of the planets, some of which had moons circling them. Planets rotated and followed an orbit around the sun, pulling their moons with them. The further a planet was from the sun, the longer it took them to make a complete orbit, or “year”. So Mars took around two years to orbit the sun, Jupiter, five years and so on to Pluto which took over thirty years to circle the sun. Even planets as far out as Pluto was subject to the great gravity of the sun.

My even deeper understanding of this powerful idea was greater than that of the teacher’s when I continued to explain that the Sun was also hurtling through space, orbiting the center of our galaxy, and that our entire galaxy was rotating and moving through space as well. The kids and the teacher agreed that I was wrong—the sun stayed still—maybe it rotated (it does) but it didn’t go anywhere and pull all those planets along. When I brought in the book that illustrated the movements of the planets, stars, and galaxies, they all basically told me to shut up and sit down!

Conceptualizing something, being able to form a mental picture, worked for me. It was somehow much more concrete and easy to understand. Much later I would be thrilled to watch an experimental film called “Cosmic Zoom”. It starts with a man lying on his back in a field, and slowly zooms out past the moon, the sun, the outer planets, the galaxy, into the furthest reaches of space—then zooming in again rapidly until the image appears of the man in the field. The camera continues to zoom into the man’s skin cells, and brings us to the atomic and molecular level before pulling back to show the man waking from his nap—oblivious to the huge cosmic dimensions that we are a part of.

Much later, when I began to use film making as a way to teach dyslexic kids and began to understand the nature of learning disabilities I thought that maybe I had a learning disability in math. Why didn’t I become an astronomer? I was repeatedly reminded that astronomers needed very powerful math skills, which I lacked, and the more I hated math, the further I got from being an astronomer.

I did overcome my later failing of algebra (grade ten, repeated in grade eleven barely passing with a D) later in college by taking physics. We did stuff like measure the speed of sound, figure out vectors, parabolic curves, and Galileo’s alleged experiment dropping cannon balls from the Tower of Pisa. We dropped them from the roof of the Physics lab. I also did extremely well in geometry my senior year of high school, because I could see it and had a good memory for logic. Using algebra to solve geometric problems was also a snap for me in geometry.

In sum, for many the teaching and learning of abstract reasoning and thinking has got to be made concrete and conceptualized either visually and/or kinesthetically. Math should be taught along multiple pathways to learning.

This is where I agree with Papert. Computers are a great tool to discover abstract reasoning through the trial and error process. Over the past few days I have been playing with microworlds logo, each attempt getting a bit better in constructing an animated model of the planets orbiting the sun. It’s still crude, and wrong, yet there are probably ways for me to program the turtles (planets) to follow ellipses—maybe even a way to have the moon orbit the earth and follow the sun! If not, I’m learning something along the way. My guess is that he would agree that computers are very misused in both our society and in most of our schools. There may be learning going on, but how much of it is constructionist?