Our in class activities in Modern Geometry as well as several of the assigned readings have heightened my awareness of the absence of conjecturing in the geometry curriculum that I present to my students. In general, my students read or are presented with the postulates and theorems that are important to a topic, and then we work with them. Very rarely are they presented with a relationship that they are asked to uncover on their own.
In their article Geometry and Proof, Battista and Clements (1995) suggest that students own conjectures are an ideal segue to the process of developing proofs. Providing students with an opportunity to discover a geometric relationship, and then leading them to defend their discoveries through class discussion and debate can demonstrate the importance of proof to students before they are asked to engage in it. The process of developing and testing a conjecture before proving it is also more true to real mathematical development. The article left me wanting to completely restructure the way proofs are presented in my geometry course, with ample time for exploration leading into informal proofs, and eventually some practice with formal deductive proofs. A progression like this would be more in line with Van Hiele‘s levels of geometric reasoning as well. Letting conjecture be the impetus for proof would make the learning of proof more intrinsic for students, and it would also make their learning a more genuine mathematical experience.
The value of conjecture in geometry courses stood out to me again in Diemente’s (2000) article about the Euler line and its relationship to the sides of a triangle. The mathematics presented in the article are certainly interesting, but I was most struck by the fact that the content of the article was all prompted by students’ questions and conjectures. Taking the time to explore a students’ question about whether the Euler line can be parallel to a side of a triangle, Diemente and his class covered a wide swath of mathematical ground. Their exploration and proof included midpoints, slopes, systems of linear equations, and even conic sections. What a great example of how valuable students’ curiosity can be! Not only were these and other concepts used, reviewed, and learned, but they were shown in connection with one another, and moreover, with a real purpose! While an exploration like that of Diemente’s honors geometry class might be over the ability levels of my geometry students, it might be a great study for students in my precalculus class.
So now I have a heightened sense of the value of providing students opportunities to make and investigate conjectures, and at the same time a greater awareness of the absence of such opportunities for my students. So now I am looking for places to create these opportunities. There are a couple on the horizon.
Tomorrow we will begin our unit on triangle congruence. Our department has an activity designed to help students discover the “shortcut” congruence theorems, like Side-Angle-Side, etc. It’s called “Triangle in a Bag”. Students can ask for one side length or angle measure of a certain triangle at a time, and the goal is to construct a triangle congruent to the one “in the bag” with as few given pieces as possible. The hope is that after doing this for several triangles, they might start to think critically about what key measurements are necessary to guarantee the same triangle, and develop the shortcut theorems. It’s a nice activity, but at the same time, I’m not sure if there’s much learning in it. A good class discussion might need to follow it, with some time for students to defend their conjectures as to why each set of three measures is enough. Any suggestions out there to make this activity better?
The chapter following this one includes the centers of triangles (incenter, circumcenter, centroid, etc.). For the first time, our department is planning to get students working with this with the aid of GeoGebra. I am looking forward to seeing how students respond, and I hope some of them will get excited about what they’re learning. Depending on how much time is available, it would be great to have students or groups of students share what they find about the relationships between these centers, or just the centers themselves. This might also be a good opportunity for some informal/formal proof writing.
I took my class to the computer lab for the first time this past week just to do some basic GeoGebra constructions and find their way around the program. They liked it, and more than once I ran across students experimenting with lines and figures on the program - being curious. I hope I can find more opportunities to let them be curious about geometry and learn through their curiosity. For that matter, I hope I can do that for all of my math students, and not just my geometry students.
Battista, M.T., & Clements, D.H. (1995). Geometry and proof. The Mathematics Teacher, 88(1), 48-54.
Diamente, D. (2000). Algebra in the service of geometry: Can Euler's line be parallel to a side of a triangle? The Mathematics Teacher, 93(5) 428-431.