An idea I’ve been interested in recently is the idea that math teachers unintentionally create misconceptions in their students by teaching explicitly a rule or principle that is useful in the short term, but will be contradicted (or even just expanded upon in some nonintuitive way) later. Teachers also sometimes create misconceptions by being unclear in an explicit explanation. Like I did, this week:
First day of two-dimensional geometry (just wrapped up angles rules and parallel lines, moving into interior and exterior angles of polygons and the Pythagorean Theorem). Class starts with some notes on polygon names, convex/concave, regular/irregular, hypotenuse, basic properties of triangles, and a few other key pieces of vocab and notation. Plan is to go through the notes, then students answer some scaffolded questions labeling, then a bunch of tougher questions building off of our last unit thinking about angle relationships in polygons.
The first page of my notes is a table–one column has the name of the polygon, one has the number of sides/angles, one has a picture of the polygon, and the final one we label the picture as convex/concave and regular/irregular. Pretty simple. Once the table got to octagon/nonagon/etc, there were no longer pictures–they just wrote down the number of sides and we moved on.
When we got to the practice, I got the same question from a bunch of kids. They know that quadrilaterals are convex and irregular, and that hexagons are concave and irregular, and heptagons are convex and regular, but what about an octagon?
I was baffled. Where did they get the idea that all quadrilaterals are convex? Or all hexagons are concave? Then I realized that they took that table as gospel. They thought that, when I wrote concave next to the sample hexagon, that I was referring to all hexagons, not just the example I had chosen.
I caught it, and fixed it in my later classes–I think a few kids in my first period class may have walked out with that misconception imprinted into their brain until they forget it over the weekend. But it was a huge waste of cognitive energy that should have been going toward some of the tougher scaffolding for the rest of our unit on polygons.
Lesson learned: structure those notes differently.
Other lesson learned: direct instruction can be really dangerous when my students are taking everything I say as truth, whether it meshes with their experience and understanding or not.