protractor photoI’ve been talking about logico-mathematical knowledge, the knowledge of relationships. Logico-mathematical knowledge is difficult to explain because it is invisible and intangible. It is the knowledge we make inside our own heads. When we make a mental connection, we often say things like “It finally clicked for me,” or “The lightbulb went on!” That’s the feeling of logico-mathematical knowledge forming in your brain.

We’ve all had the experience of trying and failing to explain something to another person. We repeat ourselves, we draw pictures, we pantomime…we try every way we can think of to convey our meaning…and still, all we get is that blank look on the other person’s face.

This shows that logico-mathematical knowledge does not come from the outside; it is not conveyed through the senses. In order for another person to understand what you are trying to say, they must already have a logical structure (what Piaget called “schema” and today we understand is a network of connections between neurons in the brain) in place in their heads. Otherwise your words are meaningless to them.

As a tutor, I encounter these knowledge disconnects all the time and I’m always on the lookout for them. They’re so subtle than most people, including most educators, miss them. But once you notice them they are startling. Here’s an example from geometry: the concept of angles.

The first time I noticed this I was flabbergasted. It was about 20 years ago, before I had begun my doctoral studies, and I was tutoring a perfectly bright and normal 10th grader in the properties of parallel lines.

I drew two parallel lines and a transversal (a third line cutting diagonally across the parallel lines). I pointed to the intersections of these lines and casually asked my student “OK, so, see the eight angles?”…

…and she replied “What angles?”

I couldn’t believe it. The angles were right there on the paper, weren’t they? Why couldn’t this girl see them? But she couldn’t! She truly did not understand what an angle was, and she had been sitting through math class after math class, confused and embarrassed, as everybody talked about angles and clearly “got it.” But she did not “get it” at all!

OK, now, consider carefully this textbook definition of an angle: An angle is formed by two rays with a common endpoint.

What, exactly, is this “thing” that is “formed”?

That common endpoint, called the vertex, is an actual point, and it is visible on the paper. The rays, also, are visible.

But the angle itself is not the vertex, and it is not the rays. The angle  is framed by the two rays. It is between the two rays.

An angle is a space. A gap. And, it is a wedge-shaped gap whose size increases the farther you get from the vertex.

Turns out that “angle” is a pretty vague and complex idea!

When I asked my student “Do you see the angle”?…there wasn’t anything for her to see, because an angle is a wedge of empty space.

For those of us who have constructed the concept of “angle” in our heads, the angle is “visible.” But until we’ve built this logico-mathematical relationship in our brains, there’s nothing for us to see when we look at an “angle.” An angle is a logico-mathematical construct which doesn’t exist in the real, visible world. It only exists within minds that have created the concept of  “angle.”

Photo by Dean Hochman