Doing math without “technology”

Math is a mental process. Basic math techniques should be, for those that are learning basic math, of the variety that you should either be able to do in your head, writing your thoughts on paper; or learn how to do so if you can’t yet. It is essentially a human skill. We acquire new math skills by building upon a foundation of what we have previously learned. In my opinion, there is nothing that modern computers can do to change this, although it might help us learn different things.

Math is a means of pure intellectual inquiry. How would you investigate the following:

A is 100 m to the north of B. A moves to the east at 3 m/s, while B moves to the west at 3 m/s. Show that the midpoint of the line AB is fixed under these conditions regardless of the position of A or B.

There are a number of ways to show this to yourself. I didn’t want to say “prove”, since I am avoiding a rigorous proof here. Just say that one rainy day, you simply asked yourself that question. You weren’t trying to be rigorous, just curious. There is no teacher in the room; no critical parents. Just you, a pencil and some paper. And maybe a ruler. And an eraser. How would you tackle it?

Some people would draw a diagram moving A and B apart in regular invervals, with \overline{AB} lines drawn between them. Others would show that the areas of the two triangles described by the initial vertical line \overline{AB} at time zero and their present position are the same area, which implies their hypoteneuses are equal. You can also show that the midpoint of the line \overline{AB} at any time > 0 is the same midpoint as exists between the initial \overline{AB} at time = 0.