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geometry, part i

December 9, 2010

Geometry is the most difficult class I teach.  It’s not the toughest of them all to learn, but it is by far the class I work hardest at perfecting.  Instilling ideas is not a simple task, especially when compared to showing people how to use rote formulas.

Take today, for example.

On Monday, we studied the Exterior Angle Theorem.  This theorem is one of many, many, many simple ways to take a diagram and end up applying algebra to solve problems, which is one of the primary goals of the course.  But even more important is to use triangles to develop some important theorems with parallel lines.  For students, this is when the class becomes a crap shoot.  Half the time, they see the connection  instantly, and half of the time they stare at the board for the rest of the class.

As a teacher, this is the worst moment of the day.

For the students who get it, further instruction is almost meaningless.  I can provide some new and interesting tangents on the concept, but it’s hardly ever needed.  They can solve any problem I can throw at them until I introduce some new and harder ideas.  If I continue to discuss the same topic, they will be bored before class ends, and will become disruptive as they demand new material.

For the students who don’t get it, they are angry and frustrated at this point.  A concept that other students clearly understand is beyond their grasp, which means they are now the intellectual inferiors in the class.  Some get upset and try harder to get it, drawing darker and darker circles in the desktops.  Others check out, claiming their inferiority and refusing to attempt understanding, instead spending their time drawing in the margins.  No matter what I discuss for the remainder of class, they will be disruptive as they wait impatiently for the end of the hour.

If I choose to teach the former group, I lose the latter, and vice versa.  If I instead attempt to teach to the middle, I lose both groups.

In many other classes, everyone can read the passage and come up with their own opinion, or write an essay, or memorize lines.  But math, and particularly geometry, is a binary game: you get it or you don’t.   I have adapted Dan Meyer’s skill-based assessment plans, and this is getting me a better idea of where my students are on this scale, but I still don’t have the faintest idea about how to get everyone in the class from 0 to 1.

There must be a better way to teach Geometry to a group of students.  I really wish I could figure out what it is.

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