“How many points define a line?” I asked the class.

 

“Three or four,” some students replied.

 

“No!” I said, “Two points define a line, and not three or four!”

 

I put a meter stick on the board, near a point. I showed them that I can rotate it, getting many lines with one point defined. Once I pick a second point, the line is defined.

 

The flag was hanging. I pointed to the stick of the flag, and told them it is a line. I then pointed to the two places where the flag was supported, as an example that two points are needed.

 

Using this, I asked what is the definition of a median line of a triangle. A student came to the board.  I drew a triangle with vertices at A, B, and C. I asked him to define a median using two points. He chose one point A. Good. He then chose the second, point B. I told him that this line is a side. He then choose the midpoint of the other side. Now he finally got it. Then I told him that point A is any vertex, and we can choose any other vertex. We get three lines. I drew them so that they do not intersect. I asked the class if this is okay. They said the lines must intersect, as this is a theorem.

 

The pedagogical principle is that to explain an idea we can show what the idea is not.

 

When we teach math, especially geometry, we must stress the logic. I repeat to my students that math is all logic. We must stress the logic of the various definitions of median lines, altitudes, etc. This logic is based upon the fact that two points define a line.