I was reminded of this story by a conversation I recently had about the purpose of teachers. I think that one of a teacher’s most important roles is as a model of a good learner. I often try to model learning for my students by asking them difficult questions that I don’t know the answer to, and working out the answer as a class, or by allowing them to pose questions and discussing their work with them. Sometimes, though, it happens accidentally. Sometimes I think or say something that’s just plain wrong – and these times can be the most educational for all. Here’s a story of a time when I accidentally put myself in a situation of not knowing and being wrong, and having to learn.
In one of my 6th grade classes, several months ago, we were investigating the sum of the measures of the angles of a polygon. I’d discovered the previous week that they had firmly entrenched in their heads that the measures of all the angles in a triangle sum to 180 degrees, and weren’t super interested in knowing why. They didn’t all seem to know that the angles of a quadrilateral sum to 360 degrees, though. I really wanted them to come up with a way of determining the angle sum of any polygon. So, I set up an investigation in which I hoped they would “discover” the angle-sum formula for polygons.
The investigation went something like this: Kids, (I asked them) for any polygon with any number of sides, what is the least number of triangles you can break that polygon into? I drew some polygons on a sheet for them to break apart and encouraged them to create their own polygons, as motivated by their investigations. This was a homework assignment.
As far as I knew, the least number of triangles you could break a polygon with n sides into was n-2. And, because the angles of a triangle sum to 180 degrees and there are n-2 triangles in an n-gon, its angle-sum is 180*(n-2). I hadn’t done a particularly large amount of research into proofs of and justifications for the angle-sum formula, but this made sense to me. It also seemed like an accessible investigation for 6th graders, with a pretty clear ending-point.
The next day, my expectations appeared to be justified. Several students reported the results I’d anticipated, and I started to go into why this showed interesting things about the angle-sum of a polygon… that is, until one kid brought up a monster. This polygon had one non-convex angle. And it had one less triangle than expected. So, according to the formula I was developing, its angle-sum was going to be 180 degrees less than other polygons with the same number of angles. Uh-oh.
No one was sure what to do, not even me. That’s right – I didn’t know what to do.
I’d like (maybe?) to say that this hardly ever happens – not knowing the answer to a mathematical question that comes up in one of my classes – but it does happen periodically, especially when I’m teaching things for the first time (which, being relatively new to the business, is often). This situation was slightly different (maybe even worse?) because not only did I not know what to do, but my original approach, on which I’d based the entire lesson, was clearly wrong. I was claiming that all polygons with the same number of sides had the same angle-sum because their least-number-of-triangles-number was the same. And here was a counter-example. Yikes!
So, here’s what I did. I said to my class of 6th graders, “Guys, I don’t know what’s going on here. I know that the formula I shared with you is correct. What about this polygon seems different from the other polygons we’ve been looking at, and how is that causing problems with this theory?”
So, we puzzled over the weird polygon together. It was a student – not me – who finally found the root of our problems. In all of the other polygons we’d studied, the vertices of the triangles all met at the vertices of the polygon. But in this goofy polygon, the concave angle was constructed from the vertex of a triangle and the edge of another triangle. So the sum of the angles of the triangles wasn’t contributing to a whole 180 degrees of the polygon angle-sum! That was where we could find the missing 180 degrees! And it explained the lack of a single triangle!
Armed with this information, we created a class version of my theory: The angle-sum of a polygon is 180*(least number of triangles you can break that polygon into), as long as the vertices of the triangles meet at the vertices of the polygon in question. Nobody has come up with a counter-example to that, yet.
I’m not proud of the lack of preparation that I put into this investigation. As an investigation in which students would uncover the angle-sum formula, it wasn’t so great. Even if the students had found that the least number of triangles you can break a polygon into is always 2 less than the number of sides, making the connection between that result and the fact that 180*(number of sides – 2) = angle-sum of a polygon was going to be a stretch for them. I was going to have to give them that result, which wasn’t my goal. So, for the future, I’m going to have to come up with a better investigation for this lesson.
But, as an investigation in which students and teacher were uncovering interesting and surprising mathematics together, it was fantastic – and was made all the better, I think, because I got stuck and needed the brain-power of my students to continue the lesson. My bafflement and intellectual excitement at being presented with that monster polygon were genuine. I was able to take part in the investigation, the thrill of asking new questions and uncovering new monsters, and the joy at having reached the result we wanted as a fellow learner, as another person who loves to do and learn mathematics. Through my not knowing, I transferred my enthusiasm to my students and helped to make the classroom a more dynamic and supportive environment for discovery. And I was able to model being a mathematician. This may have been more valuable for my students than the tighter investigation with a clearer goal that I could have constructed, with more time and thought.
By the way, if you keep looking into non-convex polygons, another interesting pattern relating the number of sides of the polygon to the least number of triangles it can be broken into appears. My 6th grade class eventually developed a formula for the least number of triangles for any polygon – convex or not – so long as the triangles don’t overlap. But, if they do overlap… Oh, the questions! Do they ever end?
I hope not.