Last week and this, one group in my class is learning about lines, rays, and measuring angles in math. One of the interesting things about this unit is that it begins from ideas that students this age understand intuitively, on a level that was taught to them years ago — lines and corners and points — and involves a shift from that straightforward comprehension to a more precise mathematical definition of those concepts. In order to understand what makes a line different from a line segment, we have to understand that lines are infinite. In order to understand what makes a point different from the dot that marks it, we have to understand that a point is merely a location with no dimension at all. It’s a shift toward understanding that our senses aren’t the basis of judgments in geometry: we can’t say two lines are parallel because they look parallel to our eyes, we have to have a way to measure and prove that they are parallel.
So, we start with a pushpin, and I ask the small group at the carpet to look at the spot at the end of the pin. We talk about how the spot the pin is pointing to is different from a circle or a sphere that we’d be able to see. We talk about the difference between a line, which is one dimensional, and a square, which is two dimensional—they know that already, so we can go farther and discuss the fact that since we can actually see the lines we draw, they aren’t mathematical lines, because they have some width on the paper. We look at the edges of pieces of paper, to get as close as we can to imagining a rectangle that’s truly two-dimensional.
And as the unit progresses over two weeks, we talk more about what we can prove. We can estimate that an angle looks like it might be around 75°, but until we learn to measure with a protractor, we can’t be sure. On the other hand, shortcuts are sometimes available: if the corner of a rectangle is split at some angle, then if we know the measurement of one side of that angle then we can subtract from 90º to find the other side. The pieces of this unit — lines and points — are in some ways very simple, but it becomes an exercise in logic, and watching the kids begin to see the ways they can work these problems out is always wonderful.