Rolfe Schmidt

I’m always amazed at how minds work. How ideas that seem completely unrelated can come together to make learning happen.

A while back, I was thinking a little bit about relativity, and had been scribbling some space-time diagrams on a legal pad. These are just 2-d plots where one axis shows where something is in “space” and the other axis shows where it is in “time”. As any object moves through space and time, you can draw a curve called a world line to show where it was when. Follow the links to see some examples. G saw this and asked what I was doing, so I tried to explain what my pictures meant. I drew a few example world lines: a walk to his grandparent’s house and back (looks like a big sideways V), mowing the lawn (a zig-zag), and sitting and watching TV (a straight line). I even drew him a little diagram to motivate the Lorentz transform, but that was just for fun.

He seemed to like it, took a legal pad, and drew about 20 little pictures that looked like space-time diagrams. It was cute.

Meanwhile, he had been struggling over a very different question. He knew that when we walked to the library and back, it was 3 miles. He wanted to know whether it was 1 mile there and 2 miles back, or 2 miles there and 1 mile back. He wasn’t happy with either alternative, but didn’t seem too receptive to my answers that it was one-and-a-half or three-halves of a mile. I tried a few different ways to convince him verbally, but failed. I broke out our fraction blocks and it seemed to work, but he still wasn’t satisfied. I didn’t have any more ideas, so I dropped it.

Then one day, we were walking to the library and he stopped and said: “Now I get it, the library is at the curve!”

I had no idea what he was talking about, so I asked him. He said: “It’s three half miles to the library because the library is at the curve!”

I still had no clue, so I told him he’d need to draw me a picture. This is what he drew:

The colored curve is the world line of our trip to the library. As you move to the right in the picture, you move forward in time. As you move up, you move away from our house. The library is the brown box at the top of the curve. Our house is the black dot at either end. The little yellow and blue things are fire hydrants we pass on our way. The brown “circles” just below the curve near the top are the local high school track. So on our walk, we start moving away from our house and forward in time, so we go up and to the right on the diagram. We stop at the library for a while and are not moving away from or toward our house, so the curve just goes to the right — still in space, forward in time. Then we come home, moving toward our house and forward in time. This is down and to the right on the diagram.

Each color represents one mile. He started breaking it into half-mile blocks below, but stopped when I told him I understood. I know, the curve is a bit too smooth and the partition into colors is a bit inaccurate (he partitioned it evenly in space-time, not in space). But he got the point.

I would never have thought to explain fractions this way. I still would never explain fractions this way to anyone other than G. I have no idea why this helped him, but it did.

The only thing I can do is keep exposing him to as many different ideas as I can.